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capm中分离定理的基本原理

capm中分离定理的基本原理

capm中分离定理的基本原理CAPM（Capital Asset Pricing Model，资本资产定价模型）中的分离定理基于马科维茨的资本组合理论（Modern Portfolio Theory，现代投资组合理论）和Harry Markowitz的财务经济学贡献。

CAPM通过分离定理表达了投资组合中风险和无风险资产的优化配置。

分离定理的基本原理包括以下关键概念：1. 有效前沿（Efficient Frontier）：在CAPM中，有效前沿是指所有可能的风险-回报组合中，能够实现最大收益（给定风险水平下）或最小风险（给定收益水平下）的投资组合集合。

2. 无差异曲线（Indifference Curve）：无差异曲线表示了投资者在风险和收益之间的权衡。

投资者在无差异曲线上的任何点上都是无差异的，即他们对于该点上的风险和回报水平持有相同的偏好。

3. 资本市场线（Capital Market Line，CML）： CML表示所有有效组合中具有最高效率的投资组合。

这条线是无风险资产（风险为零）和风险资产（整个市场投资组合）的组合线。

4. 分离定理：分离定理表达了一个关键的理念，即投资者可以通过在无风险资产和风险资产之间进行选择，实现他们在风险和回报之间的最佳权衡。

具体而言，分离定理指出，投资者可以将他们的投资决策分为两步：•选择风险系数相同的资本市场线上的一个点，该点对应于投资者的风险偏好。

•在无风险资产和选定的资本市场线上的点之间，根据他们的风险偏好选择最佳组合。

5. 均衡投资组合：在CAPM框架中，分离定理的应用使得投资者的最佳投资组合与其个人风险偏好和市场整体的风险-回报关系相关。

投资者可以通过选择均衡投资组合，将其个人风险与市场风险整合起来。

总体而言，CAPM中的分离定理提供了一种理论基础，帮助投资者理解如何在无风险资产和风险资产之间做出最优的资产配置决策。

金融学中的风险管理理论

金融学中的风险管理理论

金融学中的风险管理理论在金融领域，风险管理是一门重要的学科，其目的是通过识别、测量和控制风险，确保金融机构和投资者在面临不确定性的条件下能够保持稳健和持续的运营。

金融风险涉及到很多因素，包括市场风险、信用风险、流动性风险等。

为了有效地管理这些风险，学者们提出了不同的理论和模型。

其中最著名的风险管理理论之一是现代投资组合理论（Modern Portfolio Theory，简称MPT）。

MPT由马可维茨（Harry Markowitz）于20世纪50年代初提出，它认为投资者应该通过构建多样化的投资组合来降低风险。

具体来说，马可维茨提出了一个关于投资组合优化的模型，通过选择不同的资产组合来实现风险和收益之间的平衡。

该模型计算投资组合的预期收益和波动性，以及投资权重的最优组合，从而帮助投资者做出更明智的投资决策。

然而，MPT也存在一些局限性。

首先，它基于了一些假设，如投资者会根据预期收益和风险来评估投资组合。

但实际上，投资者往往会受到情绪和行为因素的影响，导致他们做出非理性的决策。

其次，MPT忽视了市场的非线性和非正态性特征，这意味着它在处理极端事件时可能失效。

因此，为了更准确地管理风险，学者们提出了更多的模型和理论。

另一个重要的风险管理理论是价值-at-风险（Value-at-Risk，简称VaR）。

VaR是衡量风险暴露程度的指标，它可以用来估计在给定置信水平下的最大可能损失。

通常情况下，VaR的计算基于统计方法，可以根据历史数据和概率分布来估计投资组合的风险。

然而，VaR也有一些限制。

首先，它无法提供关于损失的更多信息，只能告诉我们可能的最大损失。

其次，VaR假设市场的未来表现与历史数据相似，但实际情况往往并非如此。

因此，在使用VaR时，投资者应该考虑到其局限性，并结合其他风险管理工具来综合评估风险。

除了MPT和VaR，金融学中还有其他一些重要的风险管理理论和模型，如CAPM（Capital Asset Pricing Model，资本资产定价模型）和CVA（Credit Valuation Adjustment，信用估值调整）。

markowitz的文献综述

markowitz的文献综述

文献综述：Markowitz的资产组合理论随着金融市场的不断发展，投资者对资产配置和风险管理的需求愈发迫切。

在这个方兴未艾的环境下，哈里·马科维茨（Harry Markowitz）于1952年提出了著名的资产组合理论（Modern Portfolio Theory），该理论对资产组合和风险管理产生了深远的影响。

本文将对Markowitz的资产组合理论进行综述，探讨其核心理念、应用价值以及未来发展趋势。

一、资产组合理论的核心理念1.1 效用理论Markowitz的资产组合理论建立在效用理论的基础之上。

他提出，投资者的最终目标不是简单地追求收益最大化，而是在一定风险水平下追求效用最大化。

投资者的投资决策不仅取决于预期收益，还应考虑风险水平和资产之间的相关性。

1.2 效率前沿Markowitz将资产组合理论建模为一个多目标优化问题，他提出了“效率前沿”的概念。

效率前沿是指在给定风险水平下，投资组合所能达到的最大收益，或者在给定收益水平下，投资组合所能达到的最小风险。

通过对效率前沿的研究，投资者可以找到最优的资产配置方案。

1.3 马科维茨方差-收益均衡模型Markowitz提出了著名的方差-收益均衡模型，该模型将投资组合的风险定义为收益的方差，将投资组合的收益定义为期望收益。

他指出，投资者在选择资产配置方案时应该追求一种均衡，即在风险和收益之间取得最佳的折衷。

二、资产组合理论的应用价值2.1 风险管理Markowitz的资产组合理论为风险管理提供了重要的思路。

通过对资产之间相关性的分析和有效的风险分散，投资者可以在一定程度上规避风险，提高投资组合的抗风险能力。

2.2 盈利机会资产组合理论也为投资者提供了寻找盈利机会的方法。

通过对不同资产类别和不同资产之间相关性的分析，投资者可以发现低相关性的资产，实现有效的分散，从而获取更高的收益。

2.3 资产配置决策资产组合理论已经被广泛应用于资产配置决策中。

mpt名词解释(一)

mpt名词解释(一)

mpt名词解释(一)MPT名词解释1. MPT•MPT 是指 Modern Portfolio Theory（现代投资组合理论）的缩写。

•简要解释：MPT 是一种投资理论，旨在通过将不同类型的资产组合在一起，以最大程度地实现预期收益，同时最小化风险。

•例子：假设投资者有股票、债券和黄金这三种资产，通过 MPT，可以根据不同的风险偏好和预期收益来配置不同比例的资产，以达到最优的投资组合。

2. 风险•风险是指投资中可能面临的不确定性和可能导致损失的可能性。

•简要解释：风险是指投资项目或资产价格的波动和不确定性，它会对投资者的预期收益产生不利影响。

•例子：投资股票市场存在价格波动的风险，如果市场走势不如预期，投资者可能会面临部分或全部损失。

3. 收益率•收益率是指投资项目或资产的获利能力。

•简要解释：收益率是投资者在一定时期内获得的盈利与投资成本之间的比率。

•例子：假设某股票的购买成本为100元，一年后卖出价格为120元，那么该股票的收益率为20%。

4. 投资组合•投资组合是指将不同类型的资产以一定比例组合在一起的投资方式。

•简要解释：投资者可以通过将不同类型的资产组合在一起形成投资组合，以达到优化预期收益和控制风险的目的。

•例子：一个投资组合可以包括股票、债券、房地产等不同类型的资产，通过合理配置投资比例，以实现最佳收益和风险控制。

5. 资产配置•资产配置是指在投资组合中为不同类型的资产分配不同的权重或比例。

•简要解释：通过资产配置，投资者可以根据自身的风险承受能力和投资目标，将资金分配到不同类型的资产上，以达到最佳的投资组合。

•例子：一个投资者可以将一部分资金投资于股票，一部分投资于债券，以实现收益最大化和风险最小化的目标。

6. 标准差•标准差是衡量资产或投资组合波动性的统计工具。

•简要解释：标准差是对数据集合中数值的离散程度的一种度量。

在投资领域中，标准差可以帮助投资者衡量资产或投资组合的风险。

英文文献1

英文文献1

Improving the Markowitz Modelusing the Notion of Entropy The Mean-variance framework proposed by Markowitz is the most common model for portfolio selection problem. The most important concept in his theory is diversification. Diversification means designing an investment portfolio that reduces exposure risk by combining a variety of investments. But actually, the portfolios’ weights are often extremely concentrated on few assets when using mean-variance framework; this is a contradiction to the notion of diversification. Entropy is a well accepted measure of diversity. In this thesis, we discuss an improved mean-variance model based on maximum entropy theory (MVME). Entropy can be viewed as a measure of disparity from the uniform probability distribution. This approach can be viewed as a direct shrinkage of portfolio weights. The estimation errors, stability of portfolio weights, portfolio performance and degree of diversification for both mean-variance and the MVME framework are tested. Compared with the mean-variance framework, the improved model leads to a well diversified portfolio.Chapter 1. Introduction. Modern portfolio theory (MPT) was first discovered and developed by Harry Markowitz in his paper "Portfolio Selection," [1] published in the1952. This article presents the method to construct a portfolio that could achieve a desired level of return while minimizing the investment risk. The mean-variance framework is the most widely used model in solving portfolio diversification problems. But i t has one big weakness; the portfolios’ weights are often extremely concentrated on few assets, which is a contradiction to the notion of diversification. In this thesis, an improved model based on maximum entropy theory is discussed and we also compare it with the classical mean-variance framework. This new approach could be viewed as a combination methodology of the mean-variance and the maximum entropy theory [2].In Chapter2, the classical mean-variance framework is presented comprehensively. In Chapter 3, the conventional improvements of mean-variance framework are depicted; the properties of entropy and maximum entropy theory areintroduced. At the end of this section, the improved model based on maximum entropy theory is proposed; the parameters in MVME model and unique solution are also discussed. In Chapter 4, the comparison between mean-variance and MVME framework will be made in four aspects:• Estimation error.• Portfolio performance, including the comparison on efficient frontier, Sharp ratio,actual return and final return.• Stability of portfolio weights.• Degree of diversification.1.Introduction of Markowitz Portfolio Theory.Modern portfolio theory (MPT) is an attempt to find the balance relation of the risk-reward in the investment portfolios. MPT proposes the idea of diversification as a tool to optimize the portfolios.This theory was first discovered and developed by Harry Markowitz in the 1950’s. Markowitz showed the benefits of diversification, also known as “not putting all of your eggs in one basket” in this theory. In other words, investment is not only about picking stocks, but also about choosing the right combination of stocks. His theory emphasized the importance of risk, correlation and diversification on expected investment portfolio returns. His work changed the way that people invest.Before Markowitz, people thought that there was one optimal portfolio which could offer the maximum expected return while minimizing risk. Markowitz clarified that it is impossible from the mathematic point of view. In the real world, the optimal portfolio selection is the problem about how much should be invested in each security to achieve a desired level of return while minimizing investment risk or getting the maximum expected return at a fixed risk level. Markowitz offered an answer by the Efficient Frontier. It is possible to construct a portfolio in the “efficient frontier” to offer the maximum return for any given level of risk. Based on the above concept, Markowitz developed the famous financial portfolio model Mean-Variance model (MV model), which was published in << Portfolio Selection >> in 1952. This model is the most common formulation of the portfolio selection problem. Themean-variance analysis provides the first quantitative treatment of the tradeoff between reward and risk. As we know, the two most important factors to be considered in Markowitz portfolio selection theory are reward and risk. A fundamental question is how to measure risk. In the MV model, reward is defined by expected return while the risk is defined by variance. 2.2 Assumptions of Mean-Variance Analysis.The mean-variance analysis is based on the following assumptions [3]:1). Investors are rational and behave in a manner as to maximize their utility with a given level of income or money.2). Investors have free access to fair and correct information on the returns and the investment risk. Each investor could master the information sufficiently.3). The markets are efficient and absorb the information quickly and perfectly.4) All investors are risk-averse and try to minimize the risk and maximize return. It means that for some assets which offer the same return, the investors will prefer the lower risky one or for that level of risk an alternative portfolio which has higher expected returns exists.5). Investors make decisions based on expected returns and variance or standard deviation of these returns. Investors will accept increased risk only if compensated by higher expected rewards. Conversely, an investor who wants to seek higher returns must accept more risk.6). The returns of the investment security are random variables with a known multivariate normal distribution. With this assumption, portfolio efficiency is determined by simply compounding the expected returns and the standard deviations of their expected returns. For building up the efficient set of portfolio, as laid down by Markowitz, we need to look into these important parameters [4]:2. Rate of return.The rate of return of the asset is defined by r , satisfying that 0)1(X r X T +=where 0X and T X are the prices of the asset at purchase and selling respectively. As an example, the rate of return from deposits in a bank account is theinterest rate.3. Expected returnThe rewards of an investment in an asset have some level of uncertainty. The value of X T is unknown at time 0, which means the rates of returns are often not known in advance. We consider the rate of return as random variables. To characterize the asset we shall consider the expected rate of return. In the MV ramework, we estimate the expected value i μfor asset i as follows:∑=====Nt i i i i N t t r N r E r 1....1,1)(μ The estimated expected return is a useful way to describe the assets and gives us a generalmeasurement of how large the return it is.Variance and Standard deviation To characterize the uncertainty of an asset, we usually use the variance or standard deviation of the historical returns. It quantifies how much the rate of return deviates from the expected rate of return. The variance is defined as the risk measurement in MV framework. The estimated variance and standard deviation for asset i is given by: i Nt i it i it i sdi and u r N u r E σσ=-=-=∑=1222)(1))(( 4. Covariance between two assetsIn choosing an investment, one natural way to reduce the risk of losing value for an asset when a given event occurs is to find another asset with increasing valuewhen this event occurs. So we should not only take into account the individual returns of assets but also consider the relationship of the returns among the assets. We use the covariance to exhibit the way asset returns move together or move inversely. The covariance between asset i and j is defined as follows,))((1)))(((j jt i it j jt i it ji ij u r u r Nu r u r E --=--==σσ We note that ji ij σσ= when i ≠ j and 2i ii σσ=when i = j .If the return of asset i and j move in the same direction, we have ij σ>0 , inversely,ij σ< 0 . To describe therelation of n possible assets, we define the covariance matrix as follows: From the expression and the character of covariance, we know the covariance matrix C is symmetric and it also can be proved that matrix C is positive definite.5.Investment weightsAssume that the investor wants to select a portfolio from n possible assets, iωis the proportion invested in asset i . So if all wealth is invested, we have∑==nii11ω.The situation that a weight iωis negative corresponds to a short selling ofthe asset which means that the investor buys the asset and sells it to someone else, and uses the amount received to invest in other assets. When short selling is not allowed, we require that .6.Background.As we mentioned before, the Markowitz mean-variance framework is the most common model for solving portfolio selection problems. The most important concept in his theory is diversification. Diversification means designing an investment portfolio that reduces exposure risk by combining a variety of investments. The goal of diversification is to reduce the risk in a portfolio. However, the portfolios’ weights obtained from mean-variance framework are often extremely concentrated on a few assets. This is a contradiction to the notion of diversification. In practice, sample mean and covariance matrix are estimated from historical data. There are lots of factors that influence the estimation, such as the sample size. If the sample size is too small, the sample mean and covariance could have large estimation errors. It is generally thought that the concentrated position problem is caused by the statistical errors when estimating the mean and covariance matrix. Jobson and Korkie [8] showed that these statistical errors change the portfolio weights in such a way that often leads to that the portfolios’ weights are concentrated on some positions. And we also know that the mean-variance framework is extremely sensitive to input parameters. Small changes of the sample mean and covariance matrix will have a large effect on the optimal portfolios. So the precise estimation of sample mean andcovariance matrix is the most important prerequisite for the mean-variance framework. The method for reducing statistical errors in sample mean and covariance matrix has been widely researched. References showed that in order to reduce the statistical errors in mean-variance model, we should improve the estimation of the sample mean at least. Three different approaches may carry a good effect on estimation errors for the mean-variance model. Two of them are shrinkage estimators of sample means and the other is the bootstrap approach. So called “shrinkage” estimator i s intended to shrink the historical means to some grand mean. Consider ),(.....21T r r r R as a N×T matrix, where the rows are the time series of historical returns for each asset, the columns are the returns of different assets at a specific time. The first shrinkage estimator used to improve the sample means is called the James-Stein estimator [9]. The difference between these two shrinkage estimators is that they shrink the sample means to different targets. In the first case, the target is the arithmetic average of sample means, while the target is the mean of the MVP portfolio in sample in the second case. But we cannot say which shrinkage estimator is better in general. And The third method is the bootstrap approach [10].The bootstrap means using the resampling method to replace the actual data. The notion of bootstrap is to extract more information about the actual distribution of observed data by the generated bootstrap samples.These three methods introduced above reduce statistical errors in the parameter estimations. Furthermore, they may improve the diversification for mean-variance framework. In the next section, we will introduce a different concept called entropy to improve the diversification. This method could also be understood as a form of shrinkage of portfolio weights [11]. 3.3 Improve Portfolio Diversification Using Maximum Entropy Theory. In this section, the proposed portfolio optimization approach could be viewed as one alternative of Mean-Variance approach. As we mentioned before, we want to improve the concentrated position of portfolio weights in the mean-variance framework by directly shrinking the portfolio weights. We have already seen in the last section that entropy is a well accepted measure fordiversification. Due the nice property of entropy that the optimal solution obtained from maximizing entropy is closest to the uniform distribution, we want to add a shrink weights factor into mean-variance optimization model, hoping that it will lead to a well diversified portfolio.The following new approach could be viewed as a combination of model (1) and (8). The mean-variance model is sensitive to given data. On the other hand, the approach for finding maximum entropy is independent of given data. The use of entropy could be viewed as compensation to the risk part in MV model. It can thus decrease the reliance on data. This new approach not only uses given partial information obtained from the history sample efficiently, but also applies the entropy to adjust how much the portfolio is diversified.使用熵的概念改进马柯维茨模型马柯维茨提出的均值- 方差框架是最常用的投资组合模型选择问题。

什么是MPT方案

什么是MPT方案

什么是MPT方案引言MPT方案是一种优化资产配置的投资策略，由美国学者Harry Markowitz提出，被广泛应用于投资组合管理。

MPT，即Modern Portfolio Theory，简称现代投资组合理论，是一种基于统计和数学模型的分析方法，旨在最大化投资组合的预期收益，同时降低风险。

MPT原理MPT的核心原理是通过合理的资产配置来实现最佳投资组合的构建。

MPT认为，通过合理选择不同投资标的之间的相关性和风险特征，可以降低整个投资组合的风险，同时提高预期收益。

MPT以资产的收益率和风险为基础，通过计算投资组合的预期收益和风险，来评估不同资产组合的有效边界。

有效边界是指在给定风险水平下，可以获得最大预期收益的资产组合。

MPT的基本步骤1. 收集数据在进行MPT分析之前，首先需要收集各个投资标的的历史收益率数据。

这些数据用于计算投资标的的均值和方差，进而进行风险和关联性的评估。

2. 计算收益率和方差利用收集到的历史数据，计算每个投资标的的均值和方差。

均值代表预期收益，方差代表投资标的的风险程度。

3. 计算相关性矩阵通过计算各个投资标的之间的相关性，得出相关性矩阵。

相关性矩阵反映了不同投资标的之间的关联程度。

4. 构建投资组合利用收益率、方差和相关性矩阵，通过数学模型构建最优投资组合。

最优投资组合是指在给定风险水平下，可以获得最大预期收益的资产组合。

5. 评估投资组合对构建的投资组合进行评估，包括计算预期收益、风险和效用。

效用是指根据投资者的风险偏好，对投资组合的综合评价。

6. 调整投资组合根据投资者的风险偏好和市场变化，对投资组合进行调整。

调整包括优化投资比例、重新评估资产的价值和风险等。

MPT的优点和局限性优点•MPT通过理性的数学模型和统计分析，可以帮助投资者降低风险，提高预期收益。

•MPT考虑了不同投资标的之间的关联性，可以构建更加多样化和均衡的投资组合。

•MPT可以帮助投资者理解和管理投资组合的风险，提高投资决策的科学性和有效性。

各种金融投资理论概况

各种金融投资理论概况人类对于股市波动规律的认知，是一个极具挑战性的世界级难题。

迄今为止，尚没有任何一种理论和方法能够令人信服并且经得起时间检验——2000年，著名经济学家罗伯特·席勒在《非理性繁荣》一书中指出：“我们应当牢记，股市定价并未形成一门完美的科学”；2013年，瑞典皇家科学院在授予有效市场假说权威专家尤金·法玛和行为金融学教授罗伯特·席勒等人该年度诺贝尔经济学奖时指出：几乎没什么方法能准确预测未来几天或几周股市债市的走向，但也许可以通过研究对三年以上的价格进行预测。

当前，有关金融资产定价和股票市场波动逻辑的代表性理论，主要有如下几种：凯恩斯选美论、随机漫步理论(Random Walk Theory)、现代资产组合理论(MPT)、有效市场假说（EMH）、行为金融学（BF）等。

凯恩斯选美论选美论是由英国著名经济学家约翰·梅纳德·凯恩斯（John Maynard Keynes，1883－1946）创立的关于金融市场投资的理论。

凯恩斯应用人们熟悉的选美活动的规则及现象，研究和解释股票市场波动的规律，认为金融投资如同选美，投资人买入自己认为最有价值的股票并非至关重要，只有正确地预测其他投资者的可能动向，才能在投机市场中稳操胜券，并以类似击鼓传花的游戏来形容股市投资中的风险。

随机漫步理论（Random Walk Theory）1827年，苏格兰生物学家罗伯特·布朗（Robert Brown），发现水中的花粉及其它悬浮的微小颗粒不停地作不规则的曲线运动，继而把这种不可预测的自由运动，用自己的名字称之为“布朗运动”。

1959年，奥斯本(M．F．M Osborne) 以布朗运动原理作为研究视角，提出了随机漫步理论，认为股票交易中买方与卖方同样聪明机智，股票价格形成是市场对随机到来的事件信息作出的反应，现今的股价已基本反映了供求关系；股票价格的变化类似于“布朗运动”，具有随机漫步的特点，其变动路径没有任何规律可循。

对Modern-Portfolio-Theory现代投资组合理论的看法——课程论文

课程作业对Modern Portfolio Theory现代投资组合理论的看法课程名称：系别：年级：姓名：学号：目录1.现代组合投资理论的内容 (2)2.现代投资组合理论的应用 (5)3.学习体会 (6)4.参考文献 (7)1.现代组合投资理论的内容投资组合是为了避免过高风险和过低收益，根据多元化原则，选择若干资产进行搭配投资，不把所有的鸡蛋放在同一个篮子里，现代投资组合理论就是在这个思想下用分散的投资来优化投资组合，其是将概率论和线性代数的方法应用于证券投资组合的研究，探讨了不同类别的、运动方向各异的证券之间的内在相关性。

如图所示，E 代表期望（均值），即收益；σ代表标准差，即风险。

举个例子来说，有两个投资产品A 和B ，A 为低风险低收益产品，B 为高风险高收益产品，投资组合1为10%的A 和90%的B ，投资组合2为80%的A 和20%的B ，则可得出结论，投资组合1的风险大于投资组合2的风险。

根据收益与风险的关系又可以将投资组合的相关关系分为三种，分别是正相关关系、负相关关系和不完全相关关系，其图像如下图所示：σE （R i ）B.负相关关系（其可以通过组合使其风险等于0）C.不完全相关关系（现实情况基本上如此）而如果将投资产品变成各种组合的话，则会出现如下图所示的情况。

图中的区域表示所有可能的投资组合。

从上图中，我们就可以发现，当投资处在最外围的那条曲线上时，是最有效的，即在给定风险上，受益最大，在给定收益上，风险最小，它们是投资组合的最优解，所以称这条曲线为有效边界。

投资者只会选择有效边界上的投资组合进行投资，但是投资者究竟选择哪一个组合，则取决于投资者对风险的态度：如果是持厌恶风险态度的投资者，则会选择投资风险低的投资组合；如果是持偏好E （R i ）E（R i ） σσ风险态度的投资者，则会选择投资风险高的投资组合。

根据收益与风险的关系，我们又可以得到收益和风险的等效用曲线，及无差异曲线。

资产组合


不同相关系数下两种风险资产构成的可行集
收益Erp
r1 r2 r 1 2 2 2
(r 1 , 1 )
ρ =1
(r2 , 2 )
ρ =0
ρ =-1
风险σp
由图可见，所有两资产组合都通过2个点。无论相关系数 取什么值，组合曲线都向左凸出，其凸出的程度由相关系数 决定；ρ越小，凸出程度越大；当ρ＝－1，达到最大曲度； ρ越大，曲线越显得平滑；当ρ＝1时，曲线最为平滑。
同一条无差异曲线, 给投资者所提供的效用是无差异的，无差异曲线向 右上方倾斜, 高风险被其具有的高收益所弥补。对于每一个投资者,无差 异曲线位置越高，该曲线上对应证券组合给投资者提供的满意程度越高。
投资组合有效边界模型
最优资产组合位于无差异曲线I2与有效集相切的切点Ｏ处。
现代资产组合理论
意义：
（1）马科威茨首次对风险和收益这两个投资管 理中的基础性概念进行了准确的定义，从此 ，同时考虑风险和收益就作为描述合理投资 目标缺一不可的两个要件(参数)。 （2）投资组合理论关于分散投资的合理性的阐 述为基金管理业的存在提供了重要的理论依 据。
9
均值-方差分析方法
E ( Ri )Wi 收益 E ( R p ) i 1
n
风险
CovijWiW j Wi 2 i2 2 CovijWiW j
2 p i 1 j 1
i 1 *
n
n
n
由上式可知，证券组合的风险不仅决定于单个 证券的风险和投资比重，还决定于每个证券收 益的协方差或相关系数。
哈里· 马科维茨
1927年8月24日，生
于美国伊利诺伊州的芝加
哥。现任纽约市立大学巴鲁

Markowitz-Modern Portfolio Theory投资组合选择

投资组合选择亨利·马克维茨兰德公司投资组合的选择过程可以分为两个阶段：第一，具备敏锐的观察和丰富的经验，不断选择，直至找到你认为在未来会有良好表现的证券。

第二，依据对投资未来表现的看法，最终确定投资组合。

本文主要讨论分析第二阶段。

我们知道，投资者会选择那些可以或者应该得到最大限度的预期贴现率，或者预期回报的投资。

不管是作为一个需要解释的假说，或者是作为使投资者行为带来最大利润的引导，这个原则都是不被认同的。

我们接下来将会说明的是，投资者确实（或者应该）考虑某个表现良好的金融产品的预期收益以及某个不良资产的收益变动。

这条规则既可以作为投资行为的座右铭/标准/准则，也是投资行为的一种假设。

我们将依据资产组合收益的预期方差来阐明对投资组合的偏好与组合之间的几何关系。

投资组合选择的一种原则是投资者必须（或者应该）使未来收益的现值（或资本化值）最大1。

由于未来的不确定性，我们的现值必须得到预期的收益。

这种类型的规则变化是可预知的。

在希克斯之后，预期收益的概念中涵括了风险津贴2。

换句话说就是，我们可以假设一个利率，在这个利率上我们可以估计不同风险的一些证券能带来的收益。

“投资者必须使收益的贴现值最大化”，这一假说必须放弃。

如果我们忽视市场的不完全性，那么上述规则意味着，没有任何多元化投资组合比非多元化投资组合更具备低风险高收益的条件。

资产多样化不仅是可见的，也是明智的；如果一个行为规则，不论作为假说或者准则，都不能揭示多样化的优越性，就必须被抛弃。

*本文建立在作者在考尔斯委员会进行经济研究期间所做的工作基础上，得到了社会科学研究委员会的财政资助。

1. 参见,例如,J.B.威廉姆斯,投资价值理论(剑桥,马塞诸塞州:哈佛大学出版社,1938),55-75页。

2. J. R. 希克斯,价值和资本(纽约:牛津大学出版社,1939),126页。

希克斯将这个规则应用在一个厂商而不是一个投资组合。

上述规则并未表明以下几点：第一，预期收益是如何形成的；第二，不同的证券的贴现率是相同的还是不同的；第三，这些贴现率是如何决定的，或者说，是如何随时间变动的3。

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Modern Portfolio Theory: An OverviewIf you were to craft the perfect investment, you would probably want its attributes to include high returns coupled with little risk. The reality, of course, is that this kind of investment is next to impossible to find. Not surprisingly, people spend a lot of time developing methods and strategies that come close to the "perfect investment". But none is as popular, or as compelling, as modern portfolio theory (MPT). Here we look at the basic ideas behind MPT, the pros and cons of the theory, and how MPT affects the management of your portfolio.The TheoryOne of the most important and influential economic theories dealing with finance and investment, MPT was developed by Harry Markowitz and published under the title "Portfolio Selection" in the 1952 Journal of Finance. MPT says that it is not enough to look at the expected risk and return of one particular stock. By investing in more than one stock, an investor can reap the benefits of diversification - chief among them, a reduction in the riskiness of the portfolio. MPT quantifies the benefits of diversification, also known as not putting all of your eggs in one basket.For most investors, the risk they take when they buy a stock is that the return will be lower than expected. In other words, it is the deviation from the average return. Each stock has its own standard deviation from the mean, which MPT calls "risk".The risk in a portfolio of diverse individual stocks will be less than the risk inherent in holding any single one of the individual stocks (provided the risks ofthe various stocks are not directly related). Consider a portfolio that holds two risky stocks: one that pays off when it rains and another that pays off when it doesn't rain. A portfolio that contains both assets will always pay off, regardless of whether it rains or shines. Adding one risky asset to another can reduce the overall risk of an all-weather portfolio.In other words, Markowitz showed that investment is not just about picking stocks, but about choosing the right combination of stocks among which to distribute one's nest eggs.Two Kinds of RiskModern portfolio theory states that the risk for individual stock returns has two components:Systematic Risk - These are market risks that cannot be diversified away. Interest rates, recessions and wars are examples of systematic risks.Unsystematic Risk - Also known as "specific risk", this risk is specific to individual stocks and can be diversified away as you increase the number of stocks in your portfolio (see Figure 1). It represents the component of a stock's return that is not correlated with general market moves.For a well-diversified portfolio, the risk - or average deviation from the mean - of each stock contributes little to portfolio risk. Instead, it is the difference - or covariance - between individual stocks' levels of risk that determines overall portfolio risk. As a result, investors benefit from holding diversified portfolios instead of individual stocks.Figure 1The Efficient FrontierNow that we understand the benefits of diversification, the question of how to identify the best level of diversification arises. Enter the efficient frontier.For every level of return, there is one portfolio that offers the lowest possible risk, and for every level of risk, there is a portfolio that offers the highest return. These combinations can be plotted on a graph, and the resulting line is the efficient frontier. Figure 2 shows the efficient frontier for just two stocks - a high risk/high return technology stock (Google) and a low risk/low return consumer products stock (Coca Cola).Figure 2Any portfolio that lies on the upper part of the curve is efficient: it gives the maximum expected return for a given level of risk. A rational investor will only ever hold a portfolio that lies somewhere on the efficient frontier. The maximum level of risk that the investor will take on determines the position of the portfolio on the line.Modern portfolio theory takes this idea even further. It suggests that combining a stock portfolio that sits on the efficient frontier with a risk-free asset, the purchase of which is funded by borrowing, can actually increase returns beyond the efficient frontier. In other words, if you were to borrow to acquire a risk-free stock, then the remaining stock portfolio could have a riskier profile and, therefore, a higher return than you might otherwise choose.What MPT Means for YouModern portfolio theory has had a marked impact on how investors perceive risk, return and portfolio management. The theory demonstrates that portfolio diversification can reduce investment risk. In fact, modern money managers routinely follow its precepts.That being said, MPT has some shortcomings in the real world. For starters, it often requires investors to rethink notions of risk. Sometimes it demands that the investor take on a perceived risky investment (futures, for example) in order to reduce overall risk. That can be a tough sell to an investor not familiar with the benefits of sophisticated portfolio management techniques. Furthermore, MPT assumes that it is possible to select stocks whose individual performance is independent of other investments in the portfolio. But market historians have shown that there are no such instruments; in times of market stress, seemingly independent investments do, in fact, act as though they are related.Likewise, it is logical to borrow to hold a risk-free asset and increase your portfolio returns, but finding a truly risk-free asset is another matter. Government-backed bonds are presumed to be risk free, but, in reality, they are not. Securities such as gilts and U.S. Treasury bonds are free of default risk, but expectations of higher inflation and interest rate changes canboth affect their value.Then there is the question of the number of stocks required for diversification. How many is enough? Mutual funds can contain dozens and dozens of stocks. Investment guru William J. Bernstein says that even 100 stocks is not enough to diversify away unsystematic risk. By contrast, Edwin J. Elton and Martin J. Gruber, in their book "Modern Portfolio Theory And Investment Analysis" (1981), conclude that you would come very close to achieving optimal diversityafter adding the twentieth stock.ConclusionThe gist of MPT is that the market is hard to beat and that the people who beat the market are those who take above-average risk. It is also implied that these risk takers will get their comeuppance when markets turn down.Then again, investors such as Warren Buffett remind us that portfolio theory is just that - theory. At the end of the day, a portfolio's success rests on the investor's skills and the time he or she devotes to it. Sometimes it is better to pick a small number of out-of-favor investments and wait for the market to turn in your favor than to rely on market averages alone.Modern Portfolio Theory - MPTA theory on how risk-averse investors can construct portfolios in order to optimize marketrisk for expected returns, emphasizing that risk is an inherent part of higher reward. Also called portfolio theory or portfolio management theory.According to the theory, it's possible to construct an 'efficient frontier' of optimal portfolios offering the maximum possible expected return for a given level of risk. This theory was pioneered by Harry Markowitz in his paper "Portfolio Selection," published in 1952 by the Journal of Finance.There are four basic steps involved in the portfolio construction:-Security Valuation-Asset Allocation-Portfolio Optimization-Performance MeasurementThe Dangers of Over-DiversificationWe’ve all heard the financial exp erts expound on the benefits of diversification. And a personal stock portfolio must be diversified to some degree; none of us wishes to “put all our eggs in one basket” and expose ourselves to the inherentrisk of holding only one stock. But can you go too far in spreading your bet? This article will illustrate that there is such thing as having your portfolioover-diversified.What Is Diversification?When we talk about diversification in a stock portfolio, we’re referring to the attempt by the investor to reduce exposure to risk by investing in various companies across different sectors, industries or even countries. Most investment professionals agree that although diversification is no guarantee against loss, it is a prudent strategy to adopt towards your long-range financial objectives. (see The Importance of Diversification.) There are many studies demonstrating why diversification works, but this would involve delving into lengthy arcane financial formulas. Put simply, by spreading your investments across various sectors or industries with low correlation to each other, you reduce price volatility by the fact that not all industries and sectors move up and down at the same time or at the same rate. This provides for a more consistent overall portfolio performance.It's important to remember that no matter how diversified your portfolio is, your risk can never be shrunk down to zero. You can reduce risk associated with individual stocks (what academics call unsystematic risk), but there are inherent market risks (systematic risk) that affect nearly every stock. No amount of diversification can prevent that.Can We Diversify Away Unsystematic Risk?So, up until this point this article has begged the question: how many stocks should you own to be diversified but not over-diversified? It seems sensible to own five stocks rather than just one, but at what point does adding more stock to your portfolio cease to eliminate market risk?First off, we need to talk about how riskis defined. The generally accepted wayto measure risk is by looking at volatilitylevels. That is, the more sharply a stockor portfolio moves within a period oftime, the riskier that asset is. Astatistical concept called standard deviation is used to measure volatility. So, for the sake of this article you can think of standard deviation as meaning “risk”.According to the modern portfolio theory, you'd come very close to achieving optimal diversity after adding about the 20th stock. In Edwin J. Elton and Martin J. Gruber’s book “Modern Portfolio Theory and Investment Analysis”, they conclude that the average standard deviation (risk) of a portfolio of one stock was 49.2%, while increasing the number of stocks in the averagewell-balanced portfolio could reduce the portfolio’s standard deviation to a maximum of 19.2% (this number represents market risk). However, they also found that with a portfolio of 20 stocks the risk was reduced to about 20%. Therefore, the additional stocks from 20 to 1,000 only reduced the portfolio’s risk by about 0.8%, while the first 20 stocks reduced the portfolio’s risk by 29.2% (49.2%-20%).Many investors have the misguided view that risk is proportionately reduced with each additional stock in a portfolio, when in fact this couldn’t be farther from the truth. There is strong evidence that you can only reduce your risk to a certain point at which there is no further benefit from diversification.True DiversificationThe study mentioned above isn't suggesting that buying any 20 stocks equates with optimum diversification. Note from our original explanation ofdiversification that you need to buy stocks that are different from each other whether by company size, industry, sector, country, etc. Put in financial parlance, this means you are buying stocks that are uncorrelated – stocks that move in different directions during different times.As well, note that this article is only talking about diversification within your stock portfolio. A person’s overall portfolio should also diversify among different asset classes, meaning allocating a certain percentage to bonds, commodities, real estate, alternative assets and so on.Mutual FundsOwning a mutual fund that invests in 100 companies doesn't necessarily mean that you are at optimum diversification either. Many mutual funds are sector specific, so owning a telecom or health care mutual fund means you are diversified within that industry, but because of the high correlation between movements in stocks prices within an industry, you are not diversified to the extent you could be by investing across various industries and sectors. Balanced funds offer better risk protection than a sector-specific mutual fund because they own 100 or more stocks across the entire market.Many mutual fund holders also suffer from being over-diversified. Some funds, especially the larger ones, have so many assets (i.e. cash to invest) that they have to hold literally hundreds of stocks and consequently, so are you. In some cases this makes it nearly impossible for the fund to outperform indexes - the whole reason you invested in the fund and are paying the fund manager a management fee.For example, take Fidelity Magellan, the storied fund run by investing legend Peter Lynch from 1977 to 1990. Sin ce 1990 the fund’s assets have ballooned to more than $60 billion as of 2004. Now we’re not going to explicitly label thefund a closet index fund, but it makes you wonder how funds with this kind of level of assets are going to outperform the S&P 500? Not to mention the much higher MER as opposed to an index fund.ConclusionDiversification is like ice cream: most people would agree that both diversification and ice cream are "good" things. This doesn’t mean you can’t have too much of a good thing. Eat too much ice cream and you’ll end up with a stomach ache.The common consensus is that a well-balanced portfolio with approximately 20 stocks diversifies away the maximum amount of market risk. Owning additional stocks takes away the potential of big gainers significantly impacting your bottom line, as is the case with large mutual funds investing in hundreds of stocks. We leave you with the sage words of the “Oracle of Omaha”, Warren Buffett: "wide diversification is only required when investors do not understand what they are doing".Achieving Optimal Asset AllocationThe important task of appropriately allocating your available investment funds among different assets classes can seem daunting, with so many securities to choose from. Here we will illustrate what asset allocation is, its importance and how you can determine your appropriate asset mix and maintain it.What Is Asset Allocation?Asset allocation refers to the strategy of dividing your total investment portfolio among various asset classes, such as stocks, bonds and moneymarket securities. Essentially, asset allocation is an organized and effective method of diversification.To help determine which securities, asset classes and subclasses are optimal for your portfolio, let's define some briefly:∙Large-cap stock - These are shares issued by large companies witha market capitalization generally greater than $10 billion.∙Mid-cap stock - These are issued by mid-sized companies with a market cap generally between $2 billion and $10 billion.∙Small-cap stocks - These represent smaller-sized companies with a market cap of less than $2 billion. These types of equities tend tohave the highest risk due to lower liquidity.∙International securities - These types of assets are issued by foreign companies and listed on a foreign exchange. International securitiesallow an investor to diversify outside of his or her country, but theyalso have exposure to country risk - the risk that a country will not be able to honor its financial commitments.∙Emerging markets - This category represents securities from the financial markets of a developing country. Although investmentsin emerging markets offer a higher potential return, there is alsohigher risk, often due to political instability, country risk and lowerliquidity.∙Fixed-income securities - The fixed-income asset classcomprises debt securities that pay the holder a set amount of interest, periodically or at maturity, as well as the return of principal when thesecurity matures. These securities tend to have lower volatility thanequities, and have lower risk because of the steady income theyprovide. Note that though payment of income is promised by theissuer, there is a risk of default. Fixed-income securities includecorporate and government bonds.∙Money market - Money market securities are debt securities that are extremely liquid investments with maturities of less than oneyear. Treasury bills make up the majority of these types of securities.∙Real-estate investment trusts (REITs) - REITs trade similarly to equities, except the underlying asset is a share of a pool ofmortgages or properties, rather than ownership of a company.Maximizing Return While Minimizing RiskThe main goal of allocating your assets among various asset classes is to maximize return for your chosen level of risk, or stated another way, to minimize risk given a certain expected level of return. Of course to maximize return and minimize risk, you need to know the risk-return characteristics of the various asset classes. The following chart compares the risk and potential return of some of the more popular ones:As you can see, equities have the highest potential return, but also the highest risk. On the other hand, Treasury bills have the lowest risk since they are backed by the government, but they also provide the lowest potentialreturn.The chart also demonstrates that when you choose investments with higher risk, your expected returns also increase proportionately. But this is simply the result of the risk-return tradeoff. They will often have high volatility and are therefore suited for investors who have a high risk tolerance (can stomach wide fluctuations in value), and who have a longer time horizon.It's because of the risk-return tradeoff - which says you can seek high returns only if you are willing to take losses - that diversification through asset allocation is important. Since different assets have varying risks and experience different market fluctuations, proper asset allocation insulates your entire portfolio from the ups and downs of one single class of securities. So, while part of your portfolio may contain more volatile securities - which you've chosen for their potential of higher returns - the other part of your portfolio devoted to other assets remains stable. Because of the protection it offers, asset allocation is the key to maximizing returns while minimizing risk.Eeny, Meeny, Miney, Mo… Deciding What's Right for YouAs each asset class has varying levels of return for a certain risk, your risk tolerance, investment objectives, time horizon and available capital will provide the basis for the asset composition of your portfolio.To make the asset allocation process easier for clients, many investment companies create a series of model portfolios, each comprising different proportions of asset classes. These portfolios of different proportions satisfy a particular level of investor risk tolerance. In general, these model portfolios range from conservative to very aggressive:Conservative model portfolios generally allocate a large percent of the total portfolio to lower-risk securities such as fixed-income and money market securities.Your main goal with a conservative portfolio is to protect the principal value of your portfolio. As such, these models are often referred to as "capital preservation portfolios".Even if you are very conservative and prefer to avoid the stock market entirely, some exposure can help offset inflation. You could invest the equity portion in high-quality blue chip companies, or an index fund, since the goal is not to beat the market.A moderately conservative portfolio is ideal for those who wish to preserve a large portion of the portfolio’s total value, but are willing to take on a higheramount of risk to get some inflation protection.A common strategy within this risk level is called "current income". With this strategy, you chose securities that pay a high level of dividends or coupon payments.Moderately aggressive model portfolios are often referred to as "balanced portfolios" since the asset composition is divided almost equally between fixed-income securities and equities in order to provide a balance of growth and income.Since these moderately aggressive portfolios have a higher level of risk than those conservative portfolios mentioned above, select this strategy only if you have a longer time horizon (generally more than five years), and have a medium level of risk tolerance.Aggressive portfolios mainly consist of equities, so these portfolios' value tends to fluctuate widely. If you have an aggressive portfolio, your main goal is to obtain long-term growth of capital. As such the strategy of an aggressive portfolio is often called a "capital growth" strategy.To provide some diversification, investors with aggressive portfolios usually add some fixed-income securities.Very aggressive portfolios consist almost entirely of equities. As such, with a very aggressive portfolio, your main goal is aggressive capital growth over a long time horizon.Since these portfolios carry a considerable amount of risk, the value of the portfolio will vary widely in the short term.Nothing Is Set in StoneNote that the above outline of model portfolios and the associated strategies offer only a loose guideline - you can modify the proportions above to suit your own individual investment needs. How you fine tune the models above can depend on your future needs for capital and on what kind of an investor you are. For instance, if you like to research your own companies and devote time to stock picking, you will likely further divide your equities portion of your portfolio among subclasses of stocks. By doing so, you can achieve a specialized risk-return potential within one portion of your portfolio.Also, the amount of cash and equivalents, or money market instruments you place in your portfolio will depend on the amount of liquidity and safety you need. If you need investments that can be liquidated quickly or you would like to maintain the current value of your portfolio, you might want to puta larger portion of your investment portfolio in money market or short-term fixed-income securities. Those investors who do not have liquidity concerns and have a higher risk tolerance will have a small portion of their portfolio within these instruments.Maintaining Your PortfolioOnce you have chosen your portfolio investment strategy, it is important to conduct periodic portfolio reviews, as the value of the various assets within your portfolio will change, affecting the weighting of each asset class. For example, if you start with a moderately conservative portfolio, the value of the equity portion may increase significantly during the year, making your portfolio more like that of an investor practicing a balanced portfolio strategy, which is higher risk!In order to reset your portfolio back to its original state, you needto rebalance your portfolio. Rebalancing is the process of selling portions ofyour portfolio that have increased significantly, and using those funds to purchase additional units of assets that have declined slightly or increased at a lesser rate. This process is also important if your investment strategy or tolerance for risk has changed.SummaryAsset allocation is a fundamental investing principle, because it helps investors maximize profits while minimizing risk. The different asset allocation strategies described above can help any investor do this regardless of their risk tolerance and investment goals. In turn, choosing an appropriate asset allocation strategy and conducting periodic reviews will ensure you maintain your long-term investment goals and reach your desired return at the lowest amount of risk possible.。