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1、第9章资本资产定价模型 The Capital Asset Pricing Model,9.1 股票的需求与均衡价格9.2 资本资产定价模型9.3 资本资产定价模型的扩展形式9.4 资本资产定价模型与流动性,资本资产定价模型 Capital Asset Pricing Model(CAPM),The supply and demand for shares determine equilibrium prices and expected rates of return.Imagine a simple world with only two corporations:Bottom Up In
2、c.(BU)and Top Down Inc.(TD).Stock prices and market values are shown in Table 9.1.Investors can also invest in a money market fund(MMF)which yields a risk-free interest rate of 5%.,股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICES,Sigma Fund is a new actively managed mutual fund that has raised$220
3、 million to invest in the stock market.The security analysis staff of Sigma believes that neither BU nor TD will grow in the future and therefore,that each firm will pay level annual dividends for the foreseeable future.This is a useful simplifying assumption because,if a stock is expected to pay a
4、stream of level dividends,the ine derived from each share is a perpetuity.Therefore,the present value of each share often called the intrinsic value of the share equals the dividend divided by the appropriate discount rate.A summary of the report of the security analysts appears in Table 9.2.,股票的需求与
5、均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICES,股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICES,Using these data and assumptions Sigma easily generates the efficient frontier shown in Figure 9.1 and putes the optimal portfolio proportions corresponding to the tangency portfolio.These proportions,bin
6、ed with the total investment budget,yield the funds buy orders.With a budget of$220 million,Sigma wants a position in BU of$220,000,000 X 0.8070=$177,540,000,or$177,540,000/39=4,552,308 shares,and a position in TD of$220,000,000 X 0.1930=$42,460,000,which corresponds to 1,088,718 shares.,股票的需求与均衡价格D
7、EMAND FOR STOCKS AND EQUILIBRIUM PRICES,股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICES,The expected rates of return that Sigma used to derive its demand for shares of BU and TD were puted from the forecast of year-end stock prices and the current prices.If,say,a share of BU could be purchased at
8、 a lower price,Sigmas forecast of the rate of return on BU would be higher.Conversely,if BU shares were selling at a higher price,expected returns would be lower.A new expected return would result in a different optimal portfolio and a different demand for shares.,股票的需求与均衡价格DEMAND FOR STOCKS AND EQU
9、ILIBRIUM PRICES,股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICES,股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICES,Sigmas demand curve for BU stock is given by the Desired Shares column in Table 9.3 and is plotted in Figure 9.2.Notice that the demand curve for the stock slopes downward.When BUs s
10、tock price falls,Sigma will desire more shares for two reasons:(1)an ine effect-at a lower price Sigma can purchase more shares with the same budget,and(2)a substitution effect-the increased expected return at the lower price will make BU shares more attractive relative to TD shares.Notice that one
11、can desire a negative number of shares,that is,a short position.If the stock price is high enough,its expected return will be so low that the desire to sell will overwhelm diversification motives and investors will want to take a short position.Figure 9.2 shows that when the price exceeds$44,Sigma w
12、ants a short position in BU.,股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICES,股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICES,The demand curve for BU shares assumes that the price and therefore expected return of TD remain constant.A similar demand curve can be constructed for TD shares given a
13、 price for BU shares.As before,we would generate the demand for TD shares by revising Table 9.2 for various current prices of TD,leaving the price of BU unchanged.We use the revised expected returns to calculate the optimal portfolio for each possible price of TD,ultimately obtaining the demand curv
14、e shown in Figure 9.3.,资本资产定价模型是现代金融学的奠基石(风险与期望收益均衡模型)It is the equilibrium model that underlies all modern financial theory.由诸多简单假定原理来建立 Derived using principles of diversification with simplified assumptions.马克维茨,威廉夏普,林特纳和简莫辛研究和发展了资本资产定价模型。Markowitz,Sharpe,Lintner and Mossin are researchers credit
15、ed with its development.,资本资产定价模型 Capital Asset Pricing Model(CAPM),个体投资者是价格的接受者 Individual investors are price takers单周期投资期限 Single-period investment horizon投资限制在金融资产的交易 Investments are limited to traded financial assets无税负和交易成本 No taxes,and transaction costs,假设Assumptions,投资者是理性的均值-方差完善者 Investors
16、 are rational mean-variance optimizers同质期望 Homogeneous expectations 给定一系列证券的价格和无风险利率，所有投资者的证券收益的期望收益率与协方差矩阵相等，从而产生了有效率边界和一个独一无二的最优风险资产组合。这一假定也被称为 同质期望。Given a set of security prices and the risk-free interest rate,all investors use the same expected returns and covariance matrix of security returns
17、to generate the efficient frontier and the unique optimal risky portfolio.This assumption is often referred to as homogeneous expectations.对投资者来说信息是无成本的和有效的 Information is costless and available to all investors,假设 Assumptions(contd),全部投资者将持有相同的风险资产-市场组合 All investors will hold the same portfolio fo
18、r risky assets market portfolio.市场组合含有全部股票和每只股票在市场资产组合所占的比例等于它的市值占所有股票的市值 Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value.,均衡条件Resulting Equilibrium Conditions,市场的风险溢价取决于全部市场参与者的平均风险厌恶 Risk premium on the market d
19、epends on the average risk aversion of all market participants,均衡条件Resulting Equilibrium Conditions(cont.),式中2 M为市场资产组合的方差；A 为投资者风险厌恶的平均水平。请注意由于市场资产组合是最优资产组合，即风险有效地分散于资产组合中的所有股票，2 M也就是这个市场的系统风险。,个体证券的风险溢价是市场协方差的函数Risk premium on an individual security is a function of its covariance with the market
20、贝塔是用来测度股票与一起变动情况下证券收益的变动程度的。贝塔的正式定义如下：Beta measures the extent to which returns on the stock and the market move together.Formally,beta is defined as,均衡条件Resulting Equilibrium Conditions(cont.),个体证券的风险溢价是市场协方差的函数Risk premium on an individual security is a function of its covariance with the market 单
21、个证券的风险溢价等于：The risk premium on individual securities is,均衡条件Resulting Equilibrium Conditions(cont.),当我们把所有个人投资者的资产组合加总起来时，借与贷将互相抵消（这是因为每个借入者都有一个相应的贷出者与之对应），加总的风险资产组合价值等于整个经济中全部财富的价值，这就是市场资产组合。每只股票在这个资产组合中的比例等于股票的市值占所有股票市场价值的比例。资本资产定价模型认为每个投资者均有优化其资产组合的倾向，最终所有个人的资产组合会趋于一致，每种资产的权重等于它们在市场资产组合中所占的比例。The
22、 portfolios of all individual investors,lending and borrowing will cancel out(since each lender has a corresponding borrower),and the value of the aggregate risky portfolio will equal the entire wealth of the economy.This is the market portfolio,M.The proportion of each stock in this portfolio equal
23、s the market value of the stock(price per share times number of shares out-standing)divided by the sum of the market values of all stocks.5 The CAPM implies that as individuals attempt to optimize their personal portfolios,they each arrive at the same port-folio,with weights on each asset equal to t
24、hose of the market portfolio.,市场资产组合The Market Portfolio,依据前文给定的假定条件，不难看出所有的投资者均倾向于持有同样的风险资产 组合。如果所有的投资者都将马克维茨分析(假定5)应用于同样广泛的证券(假定3)，在一个相同的时期内计划他们的投资(假定2)，并且投资顺序内容也相同的话(假定6),那么他们必然会达到相同的最优风险资产组合。正如下图所示.Given the assumptions of the previous section,it is easy to see that all investors will desire to
25、hold identical risky portfolios.If all investors use identical Markowitz analysis(Assumption 5)applied to the same universe of securities(Assumption 3)for the same time horizon(Assumption 2)and use the same input list(Assumption 6),they all must arrive at the same determination of the optimal risky
26、portfolio,the portfolio on the efficient frontier identified by the tangency line from T-bills to that frontier,as in following figure.,市场资产组合The Market Portfolio,资本市场线Capital Market Line,E(r),E(rM),rf,M,资本市场线CML,m,M=市场组合Market portfoliorf=无风险率Risk free rateE(rM)-rf=市场风险溢价 Market risk premiumE(rM)-r
27、f=风险市场价格 Market price of risk=CAPM斜率Slope of the CAPM,M,市场风险溢价和斜率 Slope and Market Risk Premium,市场资产组合的均衡风险溢价，E(rM)-rf，与投资者群体的平均风险厌恶程度和市场资产组合的风险2M是成比例的。The equilibrium risk premium on the market portfolio,E(rM)-rf,will be proportional to the average degree of risk aversion of the investor population
28、 and the risk of the market portfolio.Now we can explain this result.,市场资产组合的风险溢价 The Risk Premium of the Market Portfolio,在简化了的CAPM模型经济中，无风险投资包括投资者之间的借入与贷出。任何借 入头寸必须同时有债权人的贷出头寸作为抵偿。这意味着投资者之间的净借入与净贷 出的总和为零。那么在风险资产组合上的投资比例总的来说是 100%，或 y=1。设y1，代入9-1式经整理，我们发现市场资产组合的风险溢价与风险厌恶的平均水平有关：In the simplified CA
29、PM economy,risk-free investments involve borrowing and lending among investors.Any borrowing position must be offset by the lending position of the creditor.This means that net borrowing and lending across all investors must be zero,and in consequence the average position in the risky portfolio is 1
30、00%,or y=1.Setting y=1 in equation 9.1 and rearranging,we find that the risk premium on the market portfolio is related to its variance by the average degree of risk aversion:,市场资产组合的风险溢价 The Risk Premium of the Market Portfolio,单个证券的风险益价是单个证券对市场组合风险的贡献函数 The risk premium on individual securities is
31、 a function of the individual securitys contribution to the risk of the market portfolio.单个证券的风险益价是构成市场组合资产收益协方差的函数 An individual securitys risk premium is a function of the covariance of returns with the assets that make up the market portfolio.,单个证券的收益和风险Expected Return and Risk on Individual Secu
32、rities,假定现在我们要测算通用公司股票的资产组合风险，我们用通用公司股票（GM股）同市场资产组合的协方差来刻画其对资产组合的风险贡献程度。为解释这种测算方法，先要再次阐明市场资产组合的方差是如何计算的。为此，我们按第 8章讨论过的方法将 n阶协方差矩阵各项按照从行到列的顺序分别乘以各证券在市场资产组合中的权重。Suppose,for example,that we want to gauge the portfolio risk of GM stock.We mea-sure the contribution to the risk of the overall portfolio fr
33、om holding GM stock by its covariance with the market portfolio.To see why this is so,let us look again at the way the variance of the market portfolio is calculated.To calculate the variance of the market port-folio,we use the bordered covariance matrix with the market portfolio weights,as discusse
34、d in Chapter 8.We highlight GM in this depiction of the n stocks in the market portfolio.,单个证券的收益和风险Expected Return and Risk on Individual Securities,单个证券的收益和风险Expected Return and Risk on Individual Securities,单个证券的收益和风险Expected Return and Risk on Individual Securities,通用公司股票对市场资产组合方差的贡献为：The contri
35、bution of GMs stock to the variance of the market portfolio is:,通用公司股票对市场资产组合方差的贡献度市场资产组合的收益率可以表示如下:The rate of return on the market portfolio may be written as,单个证券的收益和风险Expected Return and Risk on Individual Securities,通用公司股票与市场资产组合的协方差为:The covariance of the return on GM with the market portfolio
36、 is:,单个证券的收益和风险Expected Return and Risk on Individual Securities,测度了通用公司股票对市场方差的贡献度后，我们就可以来确定通用公司股票的合理风险溢价了。首先，我们注意到市场资产组合的风险溢价为E(rM-rf)，方差为2M，酬报与波动性比率为:Having measured the contribution of GM stock to market variance,we may determine the appropriate risk premium for GM.We note first that the market
37、portfolio has a risk premium of E(rM-rf)and a variance of 2M,for a reward-to-risk ratio of,单个证券的收益和风险Expected Return and Risk on Individual Securities,假定某位平均的投资者投资于市场资产组合的比例为 100%，现在他打算通过借入无风险贷款的方式来增加比例为小量的市场资产组合头寸。新的资产组合由以下三 部分组成：收益为 rM的原有市场资产组合头寸，收益为-rf 的无风险资产空头头寸，以及收益为rM的市场资产组合的多头头寸。总的资产组合收益为rM(r
38、M-rf)，将其期望值与最初期望值 E(rM)比较，期望收益的增加额为,单个证券的收益和风险Expected Return and Risk on Individual Securities,单个证券的收益和风险Expected Return and Risk on Individual Securities,单个证券的收益和风险Expected Return and Risk on Individual Securities,为了度量新资产组合的风险，我们重新计算资产组合的方差。新资产组合由权重为(1)的市场资产组合与权重为-的无风险资产组成，调整后的资产组合的方差为:To measure
39、the impact of the portfolio shift on risk,we pute the new value of the portfolio variance.The new portfolio has a weight of(1+)in the market and-in the risk-free asset.Therefore,the variance of the adjusted portfolio is:,单个证券的收益和风险Expected Return and Risk on Individual Securities,由于非常小，所以相比于2而言2可以忽略
40、，因而我们这里对这一项忽略不计 However,if is very small,then 2 will be negligible pared to 2,so we may ignore this term,单个证券的收益和风险Expected Return and Risk on Individual Securities,综合以上结果，增加的风险溢价与增加的风险之间的平衡，即风险的边际价格为:Summarizing these results,the trade-off between the incremental risk premium and incremental risk,r
41、eferred to as the marginal price of risk,is given by the ratio,单个证券的收益和风险Expected Return and Risk on Individual Securities,现在，作为一个替代，假定投资者用以无风险利率借入的资金 投资于通用公司股票。他的平均超额收益的增加值为:Now suppose that,instead,investors were to invest the incrementin GM stock,also financed by borrowing at the risk-free rate.T
42、he increase in mean excess return is:,单个证券的收益和风险Expected Return and Risk on Individual Securities,这一资产组合中投资于市场资产组合的资金权重为 1.0，投资于通用公司股票的资金权重为，投资于无风险资产的资金权重为-。这一资产组合的方差为:This portfolio has a weight of 1.0 in the market,in GM,and-in the risk-free asset.Its GM variance is:,单个证券的收益和风险Expected Return and
43、Risk on Individual Securities,因此，方差增加值包括两部分：通用公司股票新增头寸的方差和两倍通用公司股票与市场资产组合的协方差：The increase in variance therefore includes the variance of the incremental position in GM plus twice its covariance with the market:,单个证券的收益和风险Expected Return and Risk on Individual Securities,2忽略不计，通用公司股票的风险边际价格为 Droppin
44、g the negligible term involving 2,the marginal price of risk of GM is,单个证券的收益和风险Expected Return and Risk on Individual Securities,建立通用公司股票的风险边际价格同市场资产组合的风 险边际价格相等的等式如下：Equating the marginal price of risk of GMs stock to that of the market results in a relationship between the risk premium of GM and
45、that of the market:,单个证券的收益和风险Expected Return and Risk on Individual Securities,经调整我们得到通用公司股票的正常风险溢价：To determine the fair risk premium of GM stock,we rearrange slightly to obtain,单个证券的收益和风险Expected Return and Risk on Individual Securities,这里，Cov(rGM，rM)/2M测度的是通用公司股票对市场资产组合方差的贡献程度，这是市场资产组合方差的一个组成部分。
46、这一比率称作贝塔（beta），以表示，这样，9-6式可以写作为：The ratio Cov(rGM，rM)/2M measures the contribution of GM stock to the variance of the market portfolio as a fraction of the total variance of the market portfolio.The ratio is called beta and is denoted by.Using this measure,we can restate equation 9.6 as,单个证券的收益和风险Ex
47、pected Return and Risk on Individual Securities,上式即是 CAPM模型的最普通形式 期望收益-贝塔关系,我们对这一关系式还要做更详尽的论述。This expected returnbeta relationship is the most familiar expression of the CAPM to practitioners.We will have a lot more to say about the expected returnbeta relationship shortly.,单个证券的收益和风险Expected Retur
48、n and Risk on Individual Securities,If the expected returnbeta relationship holds for any individual asset,it must hold for any bination of assets.Suppose that some portfolio P has weight wk for stock k,where k takes on values 1,.,n.Writing out the CAPM equation 9.7 for each stock,and multiplying ea
49、ch equation by the weight of the stock in the portfolio,we obtain these equations,one for each stock:,证券市场线Security Market Line,E(r),E(rM),rf,证券市场线SML,b,bM=1.0,=COV(ri,rm)/m2证券市场线斜率Slope SML=E(rm)-rf=市场风险溢价market risk premium SML=rf+E(rm)-rf Betam=Cov(ri,rm)/sm2=sm2/sm2=1:证券的协方差风险,证券市场线关系SML Relatio
50、nships,系数。美国经济学家威廉夏普提出的风险衡量指标。用它反映资产组合波动性与市场波动性关系（在一般情况下，将某个具有一定权威性的股指（市场组合）作为测量股票值的基准）。如果值为1.1，即表明该股票波动性要比市场大盘高10，说明该股票的风险大于市场整体的风险，当然它的收益也应该大于市场收益，因此是进攻型证券。反之则是防守型股票。无风险证券的值等于零，市场组合相对于自身的值为1。,E(r i)=rf+i(E(rm)-rf)资本资产定价模型的最普通形式期望收益贝塔关系E(rm)-rf=.08 rf=.03x=1.25E(rx)=.03+1.25(.08)=.13 or 13%y=.6E(ry
