Portfolio Selection 2012

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不完全市场下考虑损失厌恶的连续时间投资组合选择

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3 7 * This research is supported by the National Basic Research Program of China (973 Program, Grant No. 2007CB814901). 1. Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, 230026 China; 2. School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China; 210046 † Corresponding author
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Introduction
Traditional portfolio selection models are generally established in the framework of expected utility theory (EUT) where there is an underlying assumption that investors are rational and risk averse when facing uncertainties[1−2] . They can obtain an optimal dynamic investment strategy by maximizing their expected utility of terminal wealth. This, however, has been controversial for being inconsistent with the way of people’s decision making in the real world. Over the past few years, considerable theories have been proposed to remedy the drawbacks of EUT, such as the (Cumulated) Prospect Theory[3−4] , Rank-dependent Utility Theory[5] and Dual Theory of Choice[6] . Perhaps the most famous one of them is the Cumulated Prospect Theory (CPT) of Tversky and Kahneman[4] . Recently, more and more researchers have taken an interest in incorporating CPT into the optimal portfolio selection model. Jin and Zhou[7] formulated a continuous-time portfolio selection model under CPT in a complete market, by employing the Choquet maximization and minimization technique. Gr¨ une and Semmler[8] studied the asset pricing with loss aversion in a production economy by using stochastic dynamic programming method. Barberis, Huang and Santos[9] investigated the asset price in an economy where investors derive direct utility not only from consumption but also from ﬂuctuations in the value of their ﬁnancial wealth. They explain the high mean, excess volatility, and predictability of stock returns, as well as their low correlation with the consumption growth. Gomes[10] analyzed the optimal portfolio allocation behavior of loss averse investors and its implications for trading volume. In a single-period setting, Bernard and Ghossoub[11] derived a closed-form expression of the optimal portfolio choice for a loss averse investor under a speciﬁc choice of the status quo with one risk-free asset and one risky asset. They conclude that the optimal holding in the risky asset for a CPT-investor is a function of a generalized Omega measure of the distribution of the excess return on the risky asset over the risk-free rate. Berkelaar et al.[12] derived the optimal dynamic investment strategy of loss averse investors by considering a very speciﬁc two-piece power utility function in a continuous-time complete market setting, whereas they did not consider the existence and uniqueness of the corresponding Lagrange multiplier. It is noteworthy that several studies have investigated the portfolio selection problem of risk averse agents in an incomplete market[2,13] . However, to the best of our knowledge there has been no report incorporating CPT into the portfolio selection in an incomplete market. This paper considers a very general S-shaped utility function featuring the reference point and the loss aversion of CPT in a continuous-time incomplete market setting. We suppose that the incompleteness of the market arises when the number of stocks is strictly less than the dimension of the underlying Brownian motion[13−14] . Employing the method of Zhang[14] , we can create a complete market from the incomplete one by reducing the dimension of the driving Brownian motion so that the traditional martingale method works. Then we solve the optimal terminal wealth and analyze the existence and uniqueness of the corresponding Lagrange multiplier. At the same time, the optimal wealth process and portfolio before horizon are obtained. A special example with a two-piece power utility and

浅谈贝塔系数

浅谈贝塔系数在现代财务和金融理论的研究中，风险被定义为不确定性，风险与投资的预期报酬紧紧地联系在一起，即通常所说的高风险高收益，低风险低收益。

风险分为系统性风险与非系统性风险。

系统性风险是指影响整个经济市场的风险（包括政治风险、自然风险、宏观经济风险等），非系统风险是指某些特定实体所具有的风险（包括经营风险、操作风险、财务风险等）。

市场往往只对系统风险给予投资回报，而不对非系统风险给予投资回报。

随着有关资本市场理论的建立和发展，经济学家们提出了一系列度量金融风险的方法，建立在CAPM基础上的贝塔(Beta，文中有时用β表示)系数就是其中一种广泛采用的风险度量标准，权益贝塔一般由对上市公司股票的市场价格进行回归统计得到的，对其卸载财务杠杆后可得到资产贝塔。

1952年，哈里﹒马克威茨(Harry M ．Markowitz)在Journal of Finance发表的文章“Portfolio Selection”中提出了均值——方差模型，开创性的利用数理统计语言描述了金融市场中投资者的行为，奠定了金融学定价模型的基础，成为现代金融理论的一个重要里程碑。

之后，它被人们广泛应用于实际投资组合决策。

在资产组合理论的前提下，威廉﹒夏普(William F．Sharpe) (1964)等人在Markowitz的基础上提出了CAPM理论，奠定了研究资本市场价格的理论框架。

之后，Fama(1970)提出了有效市场假说(Effective Market Hypothesis，EMH)，并给出了金融市场价格运动规律的实证检验思路。

而Black、Scholes(1973)以及Merton(1973)等人先后在CAPM的基础上提出了衍生金融品的定价模型，逐渐形成了现代金融系统中对资产定价领域内的研究框架结构。

资本资产定价模型(Capital Asset Pricing Model，CAPM)最早由Sharpe(1964)、Lintner(1965)、Mossin(1966)等人提出的，它是通过衡量某一种资产对风险的暴露而确定资产的价格，是一种寻求资产绝对价格的思路。

公司金融(基础篇)第12版 Marsha_on_Portfolio_Selection[3页]

JOHN AND MARSHA ON PORTFOLIO SELECTIONMinicase solution, Chapter 8Principles of Corporate Finance, 12th EditionR. A. Brealey, S. C. Myers and F. AllenJohn neglected to mention the standard deviation of the S&P 500. We will assume 16%. Recall th at stock i’s beta is just the ratio of its covariance with the market (σim) to the market variance σm2, where σm2 = .162 = .0256. For Pioneer Gypsum, β = .65 = σim/.0256, which gives a covariance of σim = .01664. The covariance also equals the correlation coefficient ρ times the product of the stock’s and market’s standard deviations σi and σm. For Pioneer, σim= ρσiσm= .01664 = ρ×.32×.16, which implies ρ = .325.Here is the 2×2 covariance matrix for the market and Pioneer.Now calculate the portfolio return r P, portfolio standard deviation σP and the Sharpe ratio for different fractions invested in the market and Pioneer. For example, suppose that the market gets 99% of investment and Pioneer 1%.r P = .99×.125 + .01×.11 = .12485σP2 = .992×.0256 + 2×.99×.01×.01664 + .012×.1024 = .0254σP= √.0254 = .1595Sharpe ratio = (r P–r f)/σP = (.12485 - .05)/.1595 = .4694It turns out that the Sharpe ratio is maximized by putting about 95% in the market and 5% in Pioneer.S&P 500 Pioneer Sharpe ratio1.0 0 .4688.99 .01 .4694.98 .02 .4698.97 .03 .4701.96 .04 .4702.95 .05 .4702.94 .06 .4699We can follow the same procedures for Global Mining. Global’s covariance is .03123 and its correlation with the market is .8125. The 2×2 covariance matrix is:Global’s return is not attractive: with a beta of 1.22, it should offer an expected rate of return of .05 + 1.22×.075 = .1415, over 14%. But John’s estimate is only 12.9%. Therefore he should sell Global. In fact he should eliminate it from his portfolio.Suppose that John’s starting portfolio matches the market and includes 0.75% in Global. Then he should sell all the Global shares and put the proceeds back into the overall market. The resulting portfolio weights are 100.75% in the market and -0.75% in Global. That is, the portfolio should “underweight” Global by -0.75% in order to reduce holdings of Global to zero. The underweight increases the Sharpe ratio from .4688to .4693:S&P 500 Global Sharpe ratio.99 .01 .46731.0 0 .46881.005 -.005 .46951.0075 -.0075 .4699The Sharpe ratio gets still better if the portfolio weight for Global is reduced below -.75%. A weight below -.75% means selling short. In order to sell short, John would have to borrow Global shares, sell them (with an obligation to repurchase and return the shares later) and invest the sale proceeds in the market. But we doubt that John is allowed to sell short from the portfolio he manages. We discuss short selling in Chapter 20.。

15篇现代金融学经典文献

现代金融学最经典的十几篇基石性文献(均值方差、MM、CAPM、APT、EMH、股权溢价、B-S、三因子模型、股权溢价)目录：[1] Portfolio Selection-Markowitz, JF1952(均值-方差模型)[2] Existence of an Equilibrium for a Competitive Economy-Arrow&Debreu,Econometrica1954(Arrow-Debreu均衡)[3] The Cost of Capital, Corporation Finance and the Theory ofInvestment-Modigliani&Miller, AER1958(MM定理)[4]Capital Asset Prices- A Theory of Market Equilibrium under Conditions ofRisk-Sharpe, JF1964(CAPM)[5] Security Prices, Risk, and Maximal Gains From Diversification-Lintner,JF1965(CAPM)[6] Equilibrium in a Capital Asset Market-Mossin, Econometrica1966(CAPM)[7] The Arbitrage Theory of Capital Asset Pricing-Ross, JET1976(APT)[8] The Pricing of Options and Corporate Liabilities-Black&Scholes, JPE1973(B-S)[9] Theory of Rational Option Pricing-Merton, Bell JEMS1973(B-S)[10] Proof that properly anticipated prices fluctuate randomly-Samuelson, IndustrialManagement Review1965(EMH)[11] Efficient Capital Markets- A Review of Theory and Empirical Work-Fama,JF1970(EMH)[12] Market efficiency, long-term returns, and behavioral finance-Fama,JFE1998(EMH)[13] Common risk factors in the returns on stocks and bonds-Fama&French,JFE1993(三因子模型)[14] Multifactor Explanations of Asset Pricing Anomalies-Fama&French, JF1996(三因子模型)[15] Anomalies- The Equity Premium Puzzle-Siegel&Thaler, JEP1997(股权溢价之谜)gubuqiu2009年5月6日。

组合保险策略发展综述

组合保险策略发展综述【摘要】在本文中，首先在资产理论的背景下给出了组合保险策略定义，以及相应的分类情况；继而在国内外研究现状部分，细致的描述了组合保险策略发展的历程，同时指出了国内外学者的具体贡献，并对比国内外相关的成果，指出了国内研究的不足。

【关键词】组合投资；组合保险策略；研究意义1 组合保险策略定义投资组合理论（portfolio theory）的创始人h.markowitz在1952年的关于组合选（portfolio selection）的论文中提出了风险可以量化的概念，认为通过多样化的分散投资可以降低非系统风险。

但分散投资不能免除系统风险，而系统风险对于投资者往往造成很大的伤害。

基于期权定价理论出现并成形，以及期权、期货等金融衍生产品相继出现，rubinstein&leland于1981年正式提出了投资组合保险策略。

投资组合保险策略是处理系统风险的有效手段，通过选择适当的策略，投资者不仅可以保证资产组合不会因为股市下跌而低于投资前所设定的最低标准，而且还具有增值潜力。

从组合保险策略的发展历程来看，组合保险策略产生于复杂的期权，但在实施过程中，出现了不少简化的保险策略。

在操作上组合保险策略大致可以分为两类，一类是运用期权定价公式所衍生出来的以期权为基础的组合保险策略（option.based portfolio insurance，obpi），其中包括欧式保护性看跌期权（european protective put）策略、复制性看跌期权（synthetic put option，spo）策略、亚式信托式看涨期权策略（asian fiduciary call）等。

另一种则是依据投资者的风险承受能力，设定一些简单的参数所形成的组合保险策略，如买入持有策略（buy and hold strategy，b&h）、固定比例组合保险（constant proportionportfolio insurance，ccpi）策略、时间不变性组合（time.invariant portfolioprotection，tipp）策略、固定组合（constant.mix，cm）、止损策略（stop.loss，sl）等等。

金融经济学讲义

金融经济学讲义参考教材：1.Chi-fu Huang and Robert H。

Litzenberger, 1988，Foundations for FinancialEconomics, Elservier Science Publishing Co。

, Inc.2.王江，2006，《金融经济学》,中国人民大学出版社3.张顺明、赵华，2010，《金融经济学》，首都经济贸易大学出版社.4.宋逢明,2006，《金融经济学导论》,高等教育出版社。

5.史树中，《金融经济学十讲》，6． Cochrane， J.H。

,2001, Asset Pricing, Princeton:Princeton University Press.数学预备知识：多元微积分、线性代数、初等概率论与数理统计考试：1.平常成绩占30%；2.期末考试占70％。

第一章引言1．金融经济学的研究内容:经济学的核心问题是资源的有效配置，而配置资源的效率主要体现在两个层面：在微观层面，配置资源关注的是经济参与者（包括个体、公司或政府)如何使用他们所拥有的资源来最优地满足他们的经济需要；在宏观层面,配置效率关注的是稀缺资源如何流向最能产生价值的地方。

资源的配置是通过在市场特别是金融市场上的交易来完成的，金融市场是交易金融要求权（financial claim）即对未来资源的要求权的场所。

因此，金融学关注的焦点是金融市场在资源配置中的作用和效率。

具体而言，它分析的是每一个市场参与者如何依赖金融市场达到资源的最优利用，以及市场如何促进资源在参与者之间进行有效配置.金融经济学旨在用经济学的一般原理和方法来分析金融问题，作为金融研究的入门，它主要侧重于提出金融所涉及的基本经济问题,主要包括：（1）个体参与者如何做出金融决策，尤其是在金融市场中的交易决策；(2)个体参与者的这些决策如何决定金融市场的整体行为,特别是金融要求权的价格；(3)这些价格如何影响资源的实际配置。

资产定价理论及其发展

• 表现在购买股票方面就是投资者对他们不了解的股票非常谨慎，反过来说就是人们对自己熟知的事件较为偏爱。
• Benartzi（2001）的例子显示可口可乐公司雇员将他们有权支配的养老金不少于 76%的比例投资到可口可乐公司股票上。
• 另外还有大量研究证明本土偏好偏差（ home bias）在全球多数投资者中广泛存在，如Strong & Xu（1999）、Coral & Markwitz（1999）等。
2.衍生品定价理论 • 期货的定价
持有成本理论延期交割费用理论
对冲压力理论
• 期权的定价
BSM模型二叉树模型
三、20世纪80年代以后兴起的行为金融学 1.市场异象 • Grossman-Stiglitz悖论 • 收益长期反转与中期惯性现象 • 期间效应 • 孪生股票价格之谜 • 收益率过度波动
究。给定不同标价情况下，同样的房屋的估价均值有很大差距。
7.表征性启发式偏差
• 启发式是指我们面对大量信息时处理的经验方法之一。表征性是指我们在评估事情时，倾向于依据事件与某些事件的靠近，而非依据概率进行判断。
• 例子：琳达谬误 • 琳达31岁，女，单身，性格外向，非常聪
明伶俐。她主修哲学，当她为学生时，时常关注周围平等和歧视的话题。
应用
• 交易——专业投资人是亏损规避者吗？ • Locke和Mann（1999）对CME场内交易
员进行分析表明，这些立场坚定的市场职业投资人也是卖出获利品种而尽可能的把亏损品种保持较长时间。这再次说明了交易者有亏损规避和处置效应的行为特征。 • 他们还发现，最好的交易员是那些亏损规避不明显的人，因此，请记住，最好的交易员卖出亏损品种而持有获利品种。
• 当你的观点受到新数据的挑战时，你有多少次会说：“不要紧，这不过是个例外罢了！”

资产配置

今天，资产配置的含义远比安东尼奥所理解的要丰富得多。简单地讲，资产配置就是在一个投资组合中选择资产的类别并确定其比例的过程。资产的类别有两种，一是实物资产，如房产、艺术品等；一是金融资产，如股票、债券、基金等。当投资者面对多种资产，考虑应该拥有多少种资产、每种资产各占多少比重时，资产配置的决策过程就开始了。由于各种资产往往有着截然不同的性质，历史统计也显示在相同的市场条件下它们并不总是同时地反应或同方向地反应，因此当某些资产的价值下降时，另外一些却在升值。马科维茨在上述具有里程碑意义的文章中，已经通过数量化方法说明，战略性地分散投资到收益模式有区别的资产中去，可以部分或全部填平在某些资产上的亏损，从而减少整个投资组合的波动性，使资产组合的收益趋于稳定。
然而，不管是站在塞万提斯的立场还是站在马克·吐温的阵营，资产配置的投资手段绝对不是“聪明”或“愚蠢”所能一言蔽之的。迄今为止还没有一种只赚不赔的投资理论，资产配置理论也一样。即使专家们提供了众多的背道而驰的投资建议，资产配置依然会有大行其道的市场，依然会涌现无数资产配置的热诚的实践者。
摩根.斯坦利投资集团的创始人、首席投资策略师戴维.M.达斯特，就自始至终像信徒一样虔诚于资产配置。他将资产配置视为一项艺术、一门科学，并将自己数十年的实践经验精粹在《资产配置的艺术》（TheArtofAssetAllocation）一书中与无数或专业或业余的投资者分享。达斯特说：“在过去很多年来，摩根.斯坦利的资产管理人和研究策略师都强调资产配置在实现长期投资成功上的重要性。我写这本书是要向个人和专业投资者介绍在这方面的概念、工具与相关技巧。”当今的资本市场环境已远非80多年前《华尔街日报》作出其资产配置建议时的那样，新的金融工具、资产类别和投资方式已大大拓展了投资组合的选择余地，市场环境则史无前例地反复无常。幸运的是，我们依然得到了来自华尔街的投资指导，而且这一指导是适用于任何市场环境（牛市、熊市、平衡市）、任何目标投资期限、任何组合规模和具有任何经验水平的投资者的。

Portfolio selection投资组合理论

Wiley and American Finance Association are collaborating with JSTOR to digitize, preserve and extend access to The Journal of Finance.

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风险度量研究综述

风险度量研究综述上海期货交易所首席计算机专家李大鹏博士上海期货交易所博士后工作站谷艳玲博士内容摘要：本文对金融市场风险度量理论的研究进展进行了简要的评述，着重阐述以往风险度量工具的缺陷以及最新风险度量的研究进展。

1.风险度量理论研究进展“风险”（Risk）一词本身是中性的，Savage（1954），Von和Morgenstern （1953），Dreze（1974）将风险定义为对未来结果不确定性的暴露（Exposure to uncertainty）。

以此类推，金融风险就是指金融市场的交易者在金融活动中对未来结果不确定性的暴露。

风险度量实际上就是建立一个规则，使得任何一个可能的风险头寸都对应一个数值，即风险度量大小的值。

金融风险的度量就可以看成是度量市场交易者所持有头寸的不确定收益，这个收益可以是正的（获利）也可以是负的（损失）。

如果用一个随机变量表示这个不确定收益，风险度量就是从一个随机变量集合映射到实数域的范函。

用一个简单的数字来量化金融市场交易者的风险暴露，已经成为目前比较流行的风险度量方法。

世界各地的金融机构，包括投资银行、保险公司、银行信托为增强自身运作中的风险管理能力，都在寻找合适的风险度量工具。

按照风险来源的不同，金融风险主要可以分为五种类型（Marrison，2002)：市场风险1（Market risk）, 信用风险2（Credit risk）,流动性风险3（Liquidity risk），操作风险1由于市场因素(如利率，汇率，股价以及商品价格等)的波动而导致的金融交易者的资产价值变化的风险。

这些市场因素对金融交易者造成的影响可能是直接的，也可能是通过对其交易对手，供应商或者消费者所造成的间接影响。

2由于借款人或市场交易对手的违约(无法偿付或者无法按期偿付)而导致损失的可能性。

几乎所有的金融交易都涉及信用风险问题。

3由于资产流动性降低而导致的金融交易者可能损失的风险。

当金融交易者无法通过变现资产，或者无法减轻资产作为现金等价物来偿付债务时，流动性风险就会发生。

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• To see this note that the solution to the basic problem is given in the following set-up
Define the payoff of a portfolio to be x*, so its price or value is p(x*) = E(Sx*). The investor will consume c = x* + e, where e is her labour income. Thus, her problem is choose x* to Max E [u(x* + e)] subject to E(Sx*) = W (1)
So we have λ = (p(mb) − p(me) − W)/p(x∗). Thus, the optimal portfolio is x *= mb− me − [p(mb) − p(me) − W] (1+RS) (8)
where (1+RS) = xS/p(xS) = xS/E(xS2) is the return corresponding to the discount-factor
In a discrete state space, this means
max Σi πiu(xi*+ei) subject to Σiπisixi*= W The first order conditions are u’ (xi* + ei) = λsi (2)
The optimal portfolio sets marginal utility proportional to the discount factor. The optimal portfolio itself is then xi* = u’−1(λsi) − ei (4)
• We can represent the portfolio choice as one where individuals choose a set of future payoffs rather than a set of assets (which will deliver those payoffs)
The basic portfolio problem
• The individual wishes to allocate income over time to maximise utility
• She wishes to select assets to achieve an optimum time profile to payoffs and hence consumption
and we have that total wealth (W) must equal the price of the optimal portfolio of pay-offs so W = E(si λ-1/γ si-1/γ ) which implies that λ-1/γ = W/E(si1-1/γ) By substitution we obtain xi* = W (si-1/γ)/E(si1-1/γ) (6)
The larger is γ then the larger the variation in required pay-off since the individual is more risk-averse and desires less variation in utility. Note that this is a situation where there is no labour income
Note that W acts as a scaling variable only and we can write (1+Ri*) = (si-1/γ)/E(si1-1/γ) where Ri* is the portfolio return in each state (7)
We observe that the key term is si-1/γ. In states where si is low then the investor will require a high return while for those states where si is high a low return is sufficient. The optimal portfolio will deliver such a pay-off structure.
We find the Lagrange multiplier λ by satisfying the initial wealth constraint. We can think of the variable pc(wi)/π(wi) as the SDF s(wi) in each state so we can interpret equation (4) as telling us that the individual will hold more payoff in states where the contingent claim is relatively cheap, and less in states where they have substantial labour income.
Applying this to (4) we have
cb – xi* -ei = λsi
(8)
We use the result that we can represent the SDF as a projection on the pay-off space which is a pay-off xS so that xi* = λxS + proj (cb – e│X)
The above analytical framework holds for incomplete markets. However we need to recognise that, if we do not have a complete set of contingent claims , we cannot represent the portfolio as a set of contingent claims and also we do not have a unique SDF (we discuss this later). As a consequence complete hedging of the individual risk is not possible. However the solution for the optimal portfolio (4) holds.
Under quadratic utility then marginal utility is linear. Any utility function which is non-linear can be approximated by a quadratic utility function (using Taylor’s theorem)
Portfolio Selection
Returns, income and optimal choice
Structure
• The basic portfolio problem and portfolios as pay-offs • Solving for portfolios in pay-off space • Stochastic income and other influences on portfolio selection
Solving the problem
• In the presence of complete markets the solution will involve forming a portfolio of contingent claims which provide a perfect hedge for the individual
The question arises as to how we can find the optimal portfolio. We can do this by using specific utility functions and solving approximately given the set of assets which do exist. This leads us to the next stage of the analysis
(this is an orthogonality-type condition and we will discuss in the next section)
We can solve for λ. Firstly we define factor-mimicking portfolios for preference shocks and labour income: me = proj(e│X) mb = proj(cb│X) (this is the same as a linear regression to find the portfolio return which mimics the behaviour of the labour income and bliss point respectively in a least squares sense) The wealth constraint then states W = p(x*) = −λp(x*) + p(mb) − p(me)
Stating the problem
• An individual wishes to maximise Expected Utility (E(U(C)) which is a function of terminal consumption C • We keep a very simple structure : single period, single asset to illustrate the analysis • Individual chooses the amount of risky assets to deliver an optimal payoff over different states of the world. This is the same thing as an optimal portfolio