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Return and Risk

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第四章 风险衡量
第一节 风险的数学表达 对于风险,理论上还没有统一的定义。风险都是源自未来事件的不确定性,从数学角度看,它表明的是各种结果发生的可能性。在公司金融学中,研究风险是为了研究投资的风险补偿,对风险的数学度量,是以投资(资产)的实际收益率与期望收益率的离散程度来表示的。最常见的度量指标是方差([pic]或[pic])和标准差([pic])。
一、单项证券的期望和方差
将投资收益率视为一个随机变量([pic])。期望收益率是指投资前所能预期的所有可能的收益率的期望平均值,用[pic]或[pic]表示。收益率的方差或标准差表示收益率对于期望值的偏离程度,偏离程度越高,未来收益率越波动,风险也越大。 下面以股票投资的例子来说明投资收益率的这些指标是如何计算的。 【例4.1】假设股票A一年后的投资收益率会根据未来不同的经济情况而变化,具体预测情况见表4—1。
表4—1股票A一年后预期收益率情况表
|经济情况 |发生概率(P) |股票A一年后预期收益率([pic]) |
|经济繁荣 |0.5 |20% |
|经济稳定 |0.1 |5% |
|经济衰退 |0.4 |-10% |

1. 股票A的期望收益率

[pic]=0.5×20%+0.1×5%+0.4×(-10%)=6.5% 2. 股票A的方差 [pic]=0.020025 3. 股票A的标准差 [pic]=0.141510=14.15%。 上例中,假设收益率只有三种可能的情况出现,股票A的收益率是离散型分布。如果股票A的收益率连续型分布,有无限的可能结果,就需要根据连续型分布的特征,运用积分计算期望收益率以及方差和标准差。对收益率的一个经常的假设是它符合正态分布。正态分布是对称分布,其特点是只有两个特征变量,即期望值和方差(或标准差)。因此,如果我们已知股票A收益率的期望与方差并且符合正态分布,就可以知道预期收益率的所有变化情况(见图4—1)。

图4—1 股票A收益率服从正态分布 二、证券之间的协方差和相关系数 方差和标准差表示了单个股票收益率的变动程度,如果我们要研究两个证券之间互动关系,就需要了解它们之间的协方差和相关系数。仍以例子说明协方差及相关系数的涵义和计算方法。 【例4.2】 股票A和股票B的相关信息见表4—2,计算股票A、B的协方差和相关系数,分别以[pic]和[pic]表示。
|经济情况 |发生概率(P) |股票A的预期收益率([pic]) |股票B的预期收益率([pic]) |
|经济繁荣 |0.5 |20% |40% |
|经济稳定 |0.1 |5% |10% |
|经济衰退 |0.4 |-10% |-20% |

经计算得到:[pic],[pic]; [pic],[pic]; [pic], [pic]。 [pic] [pic], [pic]。 协方差为正表明两种股票的收益率变化是同方向的,若协方差为负,则表明两种股票的收益变化方向是相反的。从协方差的定义可知,股票与自身的协方差为方差。 相关系数是标准化后的协方差,取值在-1至1之间,符号与协方差相同。正相关意味着两个变量之间同向变化,负相关则代表反向变化,当相关系数为0时,两只股票之间没有关联,此时股票间收益率的变化互不干扰;当相关系数为1时,两只股票完全正相关,因此从一只股票收益率的信息可以知道另一只股票的情况。本例中,股票B的变化幅度是股票A的两倍,因此知道股票A收益率变化后,就可以了解股票B的收益率。类似地,我们称相关系数为-l时的两只股票为完全负相关。 三、投资组合的期望和方差 现实中,可供投资人选择的投资对象并非只有一种,投资人可以将资金分散投在不同的证券或资产上。由两种以上的证券或资产所构成的集合称为投资组合(以P表示)。投资组合的风险由两个因素决定: ⑴ 组合中各个证券的风险和它们之间的相互关系; ⑵ 构成比例(以[pic]表示)——分配在各项资产上的资金占资金总数的比例。 投资组合的期望收益率[pic]就是各单项资产期望收益率的加权平均,权数为该单项资产占投资组合的比重。设组合中有[pic]项资产,则 [pic]。 投资组合的方差不是各单项资产方差的简单的加权平均,还要受到各资产之间的协方差的影响,具体为: [pic], [pic]。 【例4.3】仍沿用例4.2中的股票A和股票B的数据来说明两种证券组成的资产组合的期望值和方差的计算。假定投资者有100元,其中40%投资于股票A,60%投资于股票B, 即[pic],[pic]。 [pic]=0.4×6.5%+0.6×13%=10.4% [pic]=0.051264 [pic] 股票B的期望收益率高于股票A的期望收益率,投资于两只股票的组合的期望收益率介于两者之间。同样的结果是否也适用于方差和标准差呢? [pic] =[pic] [pic] [pic] [pic]; [pic] [pic] [pic]。 可见,由两种资产组成的投资组合,当两者是完全正相关时,组合的标准差等于两种资产标准差的加权平均(如本例中的股票A和股票B)。只要相关系数小于1,组合的标准差就小于其各自标准差的加权平均,换句话说,投资组合就产生了效果。图4—2表示了两种资产组成的投资组合期望收益率与标准差之间的一般关系。

图4—2 两种资产的投资组合期望收益率与标准差关系 A、B为具有不同期望值和标准差的资产,当[pic]时,两种资产的所有组合的可能性由线段AB表示,线段上不同的点代表了组合的不同比例,如X点表示了例子中的组合,股票A占40%,股票B占60%。多数情况下,两种资产的相关系数介于-1和1之间,从图形上看,A、B所有可能的组合即是A、B间的曲线。对于每一个相关系数值,都可以求出使得组合风险最小的资产构成,即方差和标准差最小的组合,在图中以点MV表示。

第二节 投资组合的选择

本节我们主要介绍现代资产组合管理的基本观念,为后面的风险与收益理论模型打基础。现代资产组合理论是20世纪50年代由Markowitz提出的,他在1952年发表的《资产组合选择》中第一次运用了方差和期望值作为对风险和收益的度量,从而奠定了现代风险分析的基础。他将投资者的资产选择作为出发点,指出了投资的风险和收益的权衡关系,并从理论上推导出最优的投资组合,并提出了一系列重要的概念,这也是本节学习的重点。 现代资产组合理论的基本前提:投资者都是厌恶风险的,投资者只根据期望收益率和标准差两个指标进行投资判断。因此在相同标准差的情况下,投资者会选择期望收益率最高的资产。

一、有效边界与风险资产的投资组合的选择

图4—2中,点A、B之间的曲线(或直线)代表了资产A和资产B所有可能的组合,称之为可行集合(feasible set),但是厌恶风险的投资者不会选择位于MV下方的点,因为相对于同样的标准差,他们可以在MV上方的点显示的投资组合中,获得更高的期望收益。所以投资者实际只会在MV和B之间的曲线中进行组合选择,我们称在风险相同情况下期望收益率最高,期望收益率相同情况下风险最小的投资组合为有效投资组合(efficient portfolio)。在期望收益率——标准差的图形中,表示有效投资组合的曲线被称为有效边界(efficient frontier)。 图4—2只是两种风险资产的投资组合情况,如果资产的数量超过两种,从期望收益率标准差的图形上看,可行集合不再是一条线,而是布满在一个有限的区域内(如图4—3中阴影所示)。MV和B之间的曲线是有效边界。有效边界下方的点,其期望收益要小于有效边界上的点,比如X要劣于P。有效边界右方的点,其标准差要高于有效边界上的点,比如X要劣于Y。因此投资者只会在MV和B之间的曲线中选择投资。

图4—3 多种资产的投资组合的有效边界 图4-4 投资者最优风险的确定 投资者究竟会从有效边界中选取哪个组合,取决于投资者对风险和收益的权衡态度,经济学中用效用函数来表示。图4—4中投资者效用函数的无差异曲线与有效边界相切于P点,P点就是投资者最优的投资组合,是在所有可能的组合中产生最大效用的组合。如果投资者对风险厌恶程度高,就会选择风险小的投资组合,例如Y点。 二、系统性风险和非系统性风险 已经知道,投资组合的风险既受到构成组合的资产自身的风险的影响,也受到不同资产之间相关关系的影响。如果在组合中增加资产的种类数量,对组合会有什么影响呢?为了更清楚地回答这一问题,我们假设在N种资产中,每种资产的权重是相同的,各占[pic]。则 [pic] [pic] [pic] 所以, [pic] ◆ 上式中第一项代表公司的特有风险,或称非系统性风险。 ◆ 当组合中的资产数目增多时,公司特有的风险在组合中的风险越来越小,当[pic]时, [pic],非系统性风险可以通过投资组合得到完全的化解。 ◆ 投资组合中有一部分风险不能化解,我们称之为系统性风险或市场风险,这部分风险是所有资产都需要面对的,这就是式中的第二项。 ◆ N越大,投资组合的风险越趋近于各资产收益率之间协方差的平均值。对于投资人来说,最担心的就是系统性风险。

三、无风险资产和最优投资组合

前面的最优资产组合中的资产都是有风险的。现在我们在投资选择中引入无风险资产。 无风险资产是收益率确定的资产,这一确定的收益率就是无风险利率。我们进一步假定投资人都可以在资本市场按照无风险利率自由地借贷。无风险资产的存在,使得投资人摆脱了资金的束缚,对于风险偏好高的投资人,可以通过借款买入更多的风险资产;对于保守的投资人,可以多持有无风险资产,即对外贷款。 按照无风险资产的定义,其收益率([pic])不受任何因素影响,因此它与风险资产的协方差为0。如果一个投资组合是由一个无风险资产和一个风险资产组成([pic]),它的期望值和标准差都同风险资产的相应值呈线性关系。 [pic], [pic], [pic] [pic]。

图4—5引入无风险资产后的有效资本组合 在图4—5中,⑴从(0,[pic])出发连接风险资产组合可行集各点的直线都是无风险资产和风险资产的组合,例如连接到B点的线段。 ⑵如果组合是B点,表明投资人全部持有风险资产,如果在线段的端点之间,如O点,则表明投资人还持有一部分无风险资产。 ⑶M点是过(0,[pic])的直线与风险资产有效边界的切点,可以看到CM线段下方的任何直线上的组合都不如CM上的对应组合好。 ⑷M点表明投资人将自有资金100%投入风险资产。由于投资人可以借款,它可以通过借款而持有投资组合E。 在只有风险资产的条件下,投资人的有效边界是BD曲线(设B为MV,最小方差组合),当存在无风险资产时,投资人的有效边界扩大到了过切点M和(0,[pic])的直线上,这条线被称为资本市场线(Capital Market Line,CML)。 保守的投资人可能会选择C点,而偏爱风险的投资人可能会选择E点进行投资。C点和E点都是持有相同的风险资产组合,只是风险资产和无风险资产的比例不同而已。 不同的投资人对于不同证券的收益率和风险的估计可能不同,这就会导致不同的有效边界。但如果所有投资人的预期是相同的,有效边界相同,那么所有投资人都会选择M点代表的风险资产组合,不同的只是风险投资组合与无风险资产的比例。而当所有投资人持有是相同的风险资产组合时,这一组合就成为市场组合了,即包含所有风险资产,并按每个资产的占总市值中的比例作为权重。比如,假设只有两个风险资产A、B,如果每个投资人都是按照40%对60%的比例分别投资于A、B,那么资产A的市值占整个资本市场的市值即为40%,相应地资产B的市值为总体市场总值的60%。 【定义】市场证券组合是由所有证券组成的证券组合。在这个证券组合中,投资在每种证券上的比例等于它的相对市场价值,每一种证券的相对市场价值等于这种证券的总市场价值除以所有证券的总市场价值。

第三节 风险与收益理论——资本资产定价模型

传统的资本资产定价模型(Capital Asset Pricing Model,CAPM)是在Markowitz的现代资产组合管理理论基础上发展而来的,由William Sharpe(1964),John Lintner(1965)和Morrison(1965)等人在20世纪60年代提出。 CAPM的假设: ①在一期时间模型里,投资者以期望回报率和标准差作为评价证券投资组合好坏的标准。☆ ②所有的投资者都是非满足的。 ③所有的投资者都是风险厌恶者。 ④每一种证券都是无限可分的,即投资者可以购买到他想要的一份证券的任何一部分。 ⑤无税收和交易成本。 ⑥投资人可以以无风险利率自由借贷。 ⑦所有投资者的投资周期相同。 ⑧对于所有的投资者而言,无风险利率相同。 ⑨对于所有的投资者而言,信息可以无偿地自由获得 ⑩投资者有相同的预期,即他们对证券回报率的期望、方差以及相互之间的协方差的判断一致。 基于这样的假设,资本资产定价研究在市场均衡条件下,期望收益率和风险的关系。 一、CAPM的推导 传统CAPM是通过资本市场线(CML),借助市场组合概念推导出来的。 资本市场线是在以期望收益率和方差为坐标轴的图上,表示风险资产的有效组合与一种无风险资产再组合的有效组合线。 令[pic],[pic]分别表示风险资产组合的期望收益率和方差,[pic]表示无风险资产的收益率,则无风险资产与风险资产组合再组合后的新的资产组合[pic]的期望收益率和方差分别为: [pic] [pic] 因为[pic],所以[pic]。代入[pic]中得到: [pic]。 当所有投资人的预期相同时,他们将持有相同的风险资产组合,这一组合就是市场组合,所有投资者的CML将是同一条线:[pic]。 这就是市场处于均衡状态时的CML,它反映的是市场均衡条件下,有效证券组合的期望收益率和风险的关系。当风险增加时,对应的期望收益率也增加。 现在我们建立一个风险资产[pic]与市场组合M构成的新的组合A,令[pic]表示风险资产i的期望收益率,[pic]表示风险资产[pic]的收益率与市场组合收益率之间的协方差,[pic]表示市场组合的方差。 A的期望收益率和标准差分别为: [pic] [pic]

图4-6 资产i与市场组合M的组合 我们来看图4-6,直线BC为资本市场线,点M为市场组合,在允许卖空的条件下,弧线DE反映了新的组合A的预期收益率与风险的对应关系,显然市场组合M在DE上。我们知道,任意组合的对应点与无风险资产对应点的连线的斜率表示该资产单位风险所提供的预期收益率。由均衡的性质可知,当市场达到均衡时,所有投资者持有的风险资产组合都为市场组合,此时射线BMC的斜率应该是B点与弧线DE上任意一点连线的斜率中最大的,也就是说,在市场组合M基础上,无论是增加资产还是减少资产[pic]的比例,都不能得到更高的单位风险收益,即射线AMB也就是资本市场线与弧线DE相切,切点为M。 由于包含市场组合的资本市场线的斜率相同,所以我们得到: [pic] 即:[pic] [pic]
在M点处,[pic],所以上式变形后得到
[pic] 定义: [pic], 则上式变为 [pic]。 这就是均衡的资本市场条件下的资本资产定价模型。 其中[pic]表示的是市场组合的风险溢价,贝塔系数([pic])度量的是相对于市场收益率变动,个别资产收益率同时发生变动的程度,是一个标准化后的度量单个资产对市场组合方差贡献的指标,[pic]的绝对值越大,表明单个股票收益率的波动越高。 从统计上说,[pic]实际上是通过单个股票收益率对市场指数收益率的回归得到的回归系数。实践中,市场组合多以股票市场的综合指数代替。 二、资本资产定价模型的涵义 CAPM说明如下: ⑴单个证券的期望收益率由两部分组成,无风险利率以及对所承担的风险的补偿——风险溢价。 ⑵风险溢价的大小取决于[pic]值。[pic]值越高,表明单个证券的风险越高,所得到的补偿也就越高。 ⑶[pic]值度量的是单个证券的系统性风险,非系统性风险没有风险补偿。 CAPM说明了风险与收益之间的线性关系,如果用表示证券风险的[pic]作横轴,用期望收益率作纵轴,我们得到一条斜率为市场组合风险溢价的直线,称为证券市场线(Security Market Line,SML),如图4—7所示。

图4—7 证券市场线(SML) 证券市场线上的每一点代表着不同系统风险的证券,这些点所代表的收益率是该证券最少应获得的期望收益率,它是证券市场中供求平衡的产物。假设证券市场中证券X的预期收益率高出必要的收益率(图4—7中X点),投资人对证券X的需求增加,促使证X的价格升高,收益率不断下降,直至下降到SML上的[pic]点。相反地,图中Y点代表的证券收益率偏低,其价值被高估,投资人会出售该证券,供应增加导致价格的下降,收益率逐步上升,直到SML上的[pic]点。当所有的证券都调整到均衡水平时,所有证券都会落在证券市场线上,证券市场线体现了资本市场达到均衡时,不同风险的证券的必要的收益率。 三、CAPM在实际中的应用 从CAPM产生以来,由于其简单直观的特点,这种定价模型得到了广泛的应用。在实际操作中,我们应该估计三个指标;无风险利率、市场证券组合的期望收益率以及被定价证券的[pic]值。我们可以用中、长期国债利率来近似的代替无风险利率,用某种市场指标或证券市场指数的回报率来代替市场证券组合的期望收益率,证券的[pic]值可以利用历史数据,通过统计回归方法得到。 四、对资本资产定价模型的实证研究及在中国的适用性 ㈠ 实证研究结果 ⑴ 早期对CAPM的实证研究都是支持性的。例如B1ack,Jensen和Scholes(1972)以及Fama和MacBeth(1973)对1969年之前美国资本市场的数据进行检验,发现平均股票收益与[pic]之间成正相关关系。 ⑵ 但是后来,特别是20世纪80年代后的验证结果多是负面的,平均股票收益与[pic]的正相关关系在70年代以后的数据中消失了。Fama和French(1992年)使用1962—1989年之间的数据,证明如果[pic]为惟一的解释变量,CAPM的关系不存在。 ⑶ 但同时,发现许多其他因素对股票收益具有解释能力。比较有影响的是Banz 在1981年发现的规模效应。小股票,即市场价值小的公司的股票的平均收益大大高出[pic]所预测的收益值,而大规模公司的股票表现恰恰相反。Fama和French于1992年,在同时包括[pic]、规模、账面价值和市场价值的比例,以及收益价格比的测试中,发现规模、账面价值和市场价值的比例的显著性最强,解释能力远远高于[pic]。 ⑷ 虽然这些实证研究的结果对CAPM进行了否定,但是对于实证研究方法的质疑削弱了其否定的力度。例如,针对Fama与French的结果,Amimud,Christensen和Mendelson采用相同的数据,但不同的统计检验方法,得出了[pic],实际解释了收益率。还有的质疑集中在数据的采集上,认为这一结果只对某一特定的数据集合成立,不能推广成普遍结论。 ⑸ 1977年,Roll对CAPM的检验方法提出了批判。CAPM依赖于最优风险资产组合(除股票市场外,还包括不动产、外汇等其他风险资产),即市场资产组合的存在,但在现实中是不可得的,因此要用各种市场指数作为替代。所谓的各种市场指数只能是对市场资产组合的一种近似描述,是市场组合的一个子集,实证分析中会因选择不同的指数而产生很大的差异。因此,对CAPM支持的检验只能说明作为替代的某种市场指数(例如标准一普尔500指数)的有效性。同样地,即使检验表明股票的收益率与根据标准普尔500指数算出的[pic]不相关,也不能说明CAPM是错误的,也许股票的收益率与真正的市场组合是相关的。因为,真正的市场组合不可知,所以CAPM无法检验。对市场组合认识的不同,造成对[pic]估计值的不同,因此根据CAPM的实际应用可能是误导的。 ⑹ 尽管存在各种反对CAPM的声音,但是都还不能推翻CAPM,CAPM仍是一个有待检验的模型。作为风险指标,[pic]同历史收益率有正相关的关系,尽管一般实证得出的证券市场线比理论的证券市场线要平缓(即低[pic]的股票收益率要高于按CAPM计算的预期值)。 从理论上说,CAPM第一次给出了风险定价的模型,并指出只有系统性风险得到补偿。其简便性使之在实际中得到较为广泛的应用。 ㈡ 在中国的适用性 资本资产定价模型在我国的证券市场中的应用有相当的限制。从操作性看,我国股票市场发展历史较短,统计数据不充分,因此在取得合适的、准确的股票收益率和市场收益率方面有相当的难度。从市场证券市场看,还缺乏应用的条件,主要表现在: 1.我国证券市场离有效市场有相当差距 CAPM是建立在有效市场假设条件之上的。其中关键的是共同期望以及股市规模问题。共同期望的实质是信息公开化的问题,投资者只有快速、低成本地获得信息,证券价格才能反映出真实的价值。而我国目前信息披露的及时性和准确性都存在比较严重的问题。特别是虚假信息更是误导市场,扭曲了价格。此外,中国的股市规模还相对较小,特别是还存在大量非流通股,因此股票价格易被操纵,股市长期处于供需不平衡状态,所以在均衡市场条件下推出的CAPM缺乏适用性。 2.我国证券市场投资者投资观念还不成熟 CAPM假定所有投资者都运用现代投资组合管理理论进行投资选择。而中国的股市是从以个人投资者为主体发展起来的。相当多的投资者存在赌博心理,缺乏专业方面的知识。 中国股市与CAPM的成立条件相差较远,所以CAPM在中国股市的实证研究的结果基本都是否定的。主要的实证结论包括:[pic]系数不稳定,大都偏高;股票价格表现出较强的同向波动的趋势;系统性风险占股票风险的比重高。因此利用CAPM进行分散化投资的意义不大。

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...Risk and Return Essay: Mortgage Crisis of 2008 The American Dream has been a standard set centuries ago with ideas full of prosperity and success that would drive families upward in the social ladder. The American Dream has become the character by which our country is defined; therefore, it has long been a land that is desired by others living in conditions that aren’t geared toward this ideology. It has changed throughout the years as different historical marks have altered the mindset of the United States. The ability to pursue happiness outright, education, owning a business, and leaving a legacy is the pipeline for this dream that is sought not only by people in the United States, but also by those seeking to establish themselves in this land that is overflowing with honey. One of the major factors in the American Dream which hasn’t changed much over the course of time is homeownership. Homeownership is becoming an exclusive members’ club (Jones, 2014). The increase in homeownership after 2001 provided a big boom for our economy; temporarily. In 1999, Congress passed the Gramm-Leach-Bliley Act, which was also known as the Financial Services Modernization Act of 1999. This law repealed some of the Glass-Steagall Act of 1933, allowing banks, securities companies, and insurance companies to act as a combination of an investment back, commercial bank, and an insurance company which created financial supermarkets (Jenkins, 2012). The United States economy was in...

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