资本定价模型课件.ppt

9-1第9章资本资产定价模型 The Capital Asset Pricing Model9-29.1 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)9-3 The The supply supply and and demand demand for for shares shares determine determine equilibrium equilibrium prices prices and and expected expected rates rates of of return. return. Imagine Imagine a a simple simple world world with with only only two two corporations: corporations: Bottom Bottom Up Up Inc. Inc. (BU) (BU) and and Top Top Down Down Inc. Inc. (TD). (TD). Stock Stock prices prices and and market market values values are are shown shown in in Table Table 9.1. 9.1. Investors Investors can can also also invest invest in in a a money money market market fund (MMF) which yields a risk-free interest rate of 5%.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 PRICESDEMAND FOR STOCKS AND EQUILIBRIUM PRICES 9-4 Sigma Sigma Fund Fund is is a a new new actively actively managed managed mutual mutual fund fund that that has has raised raised $220 $220 million million to to invest invest in in the the stock stock market. market. The The security security analysis analysis staff staff of of Sigma Sigma believes believes that that neither neither BU BU nor nor TD TD will will grow grow in in the the future future and and therefore, therefore, that that each each firm firm will will pay pay level level annual annual dividends dividends for for the the foreseeable foreseeable future. future. This This is is a a useful useful simplifying simplifying assumption assumption because, because, if if a a stock stock is is expected expected to to pay pay a a stream stream of of level level dividends, dividends, the the income income derived derived from from each each share share is is a a perpetuity. perpetuity. Therefore, Therefore, the the present present value value of of each each share share often often called called the the intrinsic intrinsic value value of of the the share share equals equals the the dividend dividend divided divided by by the the appropriate appropriate discount discount rate. rate. A A summary summary of the report of the security analysts appears in Table 9.2.of the report of the security analysts appears in Table 9.2.股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICESDEMAND FOR STOCKS AND EQUILIBRIUM PRICES 9-5股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICESDEMAND FOR STOCKS AND EQUILIBRIUM PRICES 9-6 Using these data and assumptions Sigma easily Using these data and assumptions Sigma easily generates the efficient frontier shown in Figure 9.1 generates the efficient frontier shown in Figure 9.1 and computes the optimal portfolio proportions and computes the optimal portfolio proportions corresponding to the tangency portfolio. These corresponding to the tangency portfolio. These proportions, combined with the total investment proportions, combined with the total investment budget, yield the fund’s buy orders. With a budget of budget, yield the fund’s buy orders. With a budget of $220 million, Sigma wants a position in BU of $220 million, Sigma wants a position in BU of $220,000,000 X 0.8070 =$177,540,000, or $220,000,000 X 0.8070 =$177,540,000, or $177,540,000/39 =4,552,308 shares, and a position $177,540,000/39 =4,552,308 shares, and a position in TD of $220,000,000 X 0.1930= $42,460,000, in TD of $220,000,000 X 0.1930= $42,460,000, which corresponds to 1,088,718 shares.which corresponds to 1,088,718 shares.股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICESDEMAND FOR STOCKS AND EQUILIBRIUM PRICES 9-7股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICESDEMAND FOR STOCKS AND EQUILIBRIUM PRICES 9-8 The expected rates of return that Sigma used to The expected rates of return that Sigma used to derive its demand for shares of BU and TD were derive its demand for shares of BU and TD were computed from the forecast of year-end stock computed from the forecast of year-end stock prices and the current prices. If, say, a share of BU prices and the current prices. If, say, a share of BU could be purchased at a lower price, Sigma’s could be purchased at a lower price, Sigma’s forecast of the rate of return on BU would be forecast of the rate of return on BU would be higher. Conversely, if BU shares were selling at a higher. Conversely, if BU shares were selling at a higher price, expected returns would be lower. A higher price, expected returns would be lower. A new expected return would result in a different new expected return would result in a different optimal portfolio and a different demand for optimal portfolio and a different demand for shares.shares.股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICESDEMAND FOR STOCKS AND EQUILIBRIUM PRICES 9-9股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICESDEMAND FOR STOCKS AND EQUILIBRIUM PRICES 9-10股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICESDEMAND FOR STOCKS AND EQUILIBRIUM PRICES Sigma’s demand curve for BU stock is given by the Desired Sigma’s demand curve for BU stock is given by the Desired Shares column in Table 9.3 and is plotted in Figure 9.2. Shares column in Table 9.3 and is plotted in Figure 9.2. Notice that the demand curve for the stock slopes Notice that the demand curve for the stock slopes downward. When BU’s stock price falls, Sigma will desire downward. When BU’s stock price falls, Sigma will desire more shares for two reasons: (1) an income effect - at a more shares for two reasons: (1) an income effect - at a lower price Sigma can purchase more shares with the lower price Sigma can purchase more shares with the same budget, and (2) a substitution effect - the increased same budget, and (2) a substitution effect - the increased expected return at the lower price will make BU shares expected return at the lower price will make BU shares more attractive relative to TD shares. Notice that one can more attractive relative to TD shares. Notice that one can desire a negative number of shares, that is, a short desire a negative number of shares, that is, a short position. If the stock price is high enough, its expected position. If the stock price is high enough, its expected return will be so low that the desire to sell will overwhelm return will be so low that the desire to sell will overwhelm diversification motives and investors will want to take a diversification motives and investors will want to take a short position. Figure 9.2 shows that when the price short position. Figure 9.2 shows that when the price exceeds $44, Sigma wants a short position in BU.exceeds $44, Sigma wants a short position in BU.9-11股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICESDEMAND FOR STOCKS AND EQUILIBRIUM PRICES 9-12股票的需求与均衡价格DEMAND FOR STOCKS AND EQUILIBRIUM PRICESDEMAND FOR STOCKS AND EQUILIBRIUM PRICES The demand curve for BU shares assumes that The demand curve for BU shares assumes that the price and therefore expected return of TD the price and therefore expected return of TD remain constant. A similar demand curve can be remain constant. A similar demand curve can be constructed for TD shares given a price for BU constructed for TD shares given a price for BU shares. As before, we would generate the shares. As before, we would generate the demand for TD shares by revising Table 9.2 for demand for TD shares by revising Table 9.2 for various current prices of TD, leaving the price of various current prices of TD, leaving the price of BU unchanged. We use the revised expected BU unchanged. We use the revised expected returns to calculate the optimal portfolio for each returns to calculate the optimal portfolio for each possible price of TD, ultimately obtaining the possible price of TD, ultimately obtaining the demand curve shown in Figure 9.3.demand curve shown in Figure 9.3.9-13资本资产定价模型是现代金融学的奠基石资本资产定价模型是现代金融学的奠基石( (风险与期望风险与期望收益均衡模型收益均衡模型) ) It is the equilibrium model that underlies all modern It is the equilibrium model that underlies all modern financial theory.financial theory.由诸多简单假定原理来建立由诸多简单假定原理来建立 Derived using principles of diversification with Derived using principles of diversification with simplified assumptions.simplified assumptions.马克维茨马克维茨, , 威廉威廉· ·夏普夏普, ,林特纳和简林特纳和简· ·莫辛研究和发展了资莫辛研究和发展了资本资产定价模型。

本资产定价模型 Markowitz, Sharpe, Lintner and Mossin are Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development.researchers credited with its development.资本资产定价模型 Capital Asset Pricing Model (CAPM)9-14１个体投资者是价格的接受者１个体投资者是价格的接受者 　　　　Individual investors are price takersIndividual investors are price takers２单周期投资期限２单周期投资期限 Single-period investment horizonSingle-period investment horizon３投资限制在金融资产的交易３投资限制在金融资产的交易 Investments are limited to traded financial assetsInvestments are limited to traded financial assets４无税负和交易成本４无税负和交易成本 No taxes, and transaction costsNo taxes, and transaction costs假设Assumptions9-15５投资者是理性的均值５投资者是理性的均值- -方差完善者方差完善者 Investors are rational mean-variance optimizers Investors are rational mean-variance optimizers６同质期望６同质期望 Homogeneous expectations 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 同质期同质期望 Given a set of security prices and the risk-free interest rate, Given a set of security prices and the risk-free interest rate, all investors use the same expected returns and covariance all investors use the same expected returns and covariance matrix of security returns to generate the efficient frontier and matrix of security returns to generate the efficient frontier and the unique optimal risky portfolio. This assumption is often the unique optimal risky portfolio. This assumption is often referred to as referred to as homogeneous expectationshomogeneous expectations. .　　 对投资者来说信息是无成本的和有效的对投资者来说信息是无成本的和有效的 Information is costless and available to all investorsInformation is costless and available to all investors假设 Assumptions (cont’d)9-16全部投资者将持有相同的风险资产全部投资者将持有相同的风险资产- -市场组合市场组合 All investors will hold the same portfolio for risky All investors will hold the same portfolio for risky assets assets – – market portfolio. market portfolio.市场组合含有全部股票和每只股票在市场资产组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 Market portfolio contains all securities and the proportion of each security is its market value as proportion of each security is its market value as a percentage of total market value.a percentage of total market value.均衡条件Resulting Equilibrium Conditions9-17市场的风险溢价取决于全部市场参与者的平均风市场的风险溢价取决于全部市场参与者的平均风险厌恶险厌恶 Risk premium on the market depends on the Risk premium on the market depends on the average risk aversion of all market participantsaverage risk aversion of all market participants均衡条件Resulting Equilibrium Conditions (cont.)式中σ2 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， σ2 M也也就是这个市场的系统风险9-18个体证券的风险溢价是市场协方差的函数个体证券的风险溢价是市场协方差的函数Risk Risk premium on an individual security is a function of premium on an individual security is a function of its covariance with the marketits covariance with the market 贝塔是用来测度股票与一起变动情况下证券收益的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：变动程度的贝塔的正式定义如下： Beta measures the extent to which returns on the Beta measures the extent to which returns on the stock and the market move together. Formally, stock and the market move together. Formally, beta is defined asbeta is defined as均衡条件Resulting Equilibrium Conditions (cont.)9-19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.)9-20 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所占的比例 The portfolios of all individual investors, lending and borrowing will The portfolios of all individual investors, lending and borrowing will cancel out (since each lender has a corresponding borrower), and the cancel out (since each lender has a corresponding borrower), and the value of the aggregate risky portfolio will equal the entire wealth of the value of the aggregate risky portfolio will equal the entire wealth of the economy. This is the market portfolio, economy. This is the market portfolio, MM. The proportion of each stock in . The proportion of each stock in this portfolio equals the market value of the stock (price per share times this portfolio equals the market value of the stock (price per share times number of shares out- standing) divided by the sum of the market values 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 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 their personal portfolios, they each arrive at the same port- folio, with weights on each asset equal to those of the market portfolio.weights on each asset equal to those of the market portfolio.市场资产组合The Market Portfolio9-21 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组合如果所有的投资者都将马克维茨分析( (假定假定5)5)应用于同样广泛的证券应用于同样广泛的证券( (假定假定3)3)，在一个相同的时期内计，在一个相同的时期内计划他们的投资划他们的投资( (假定假定2)2)，并且投资顺序内容也相同的话，并且投资顺序内容也相同的话( (假定假定6),6),那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 Given the assumptions of the previous section, it is easy to see that all investors will desire to hold identical risky see that all investors will desire to hold identical risky portfolios. If all investors use identical Markowitz analysis portfolios. If all investors use identical Markowitz analysis (Assumption 5) applied to the same universe of securities (Assumption 5) applied to the same universe of securities (Assumption 3) for the same time horizon (Assumption 2) and (Assumption 3) for the same time horizon (Assumption 2) and use the same input list (Assumption 6), they all must arrive at use the same input list (Assumption 6), they all must arrive at the same determination of the optimal risky portfolio, the the same determination of the optimal risky portfolio, the portfolio on the efficient frontier identified by the tangency line portfolio on the efficient frontier identified by the tangency line from T-bills to that frontier, as in following figure. from T-bills to that frontier, as in following figure. 市场资产组合The Market Portfolio9-22资本市场线Capital Market LineE(r)E(rM)rfM资本市场线CMLm9-23MM= =市场组合市场组合Market portfolioMarket portfolior rf f= =无风险率无风险率Risk free rateRisk free rateE(rE(rMM) - r) - rf f= =市场风险溢价市场风险溢价 Market risk premiumMarket risk premiumE(rE(rMM) - r) - rf f= =风险市场价格风险市场价格 Market price of riskMarket price of risk= = CAPM CAPM斜率斜率Slope of the CAPMSlope of the CAPMM 市场风险溢价和斜率 Slope and Market Risk Premium9-24 市场资产组合的均衡风险溢价，市场资产组合的均衡风险溢价， E E( (r rM M)-)-r rf f，与投资者，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群体的平均风险厌恶程度和市场资产组合的风险σσ2 2M M是成比例的。

是成比例的 The equilibrium risk premium on the market The equilibrium risk premium on the market portfolio, E(rportfolio, E(rM M)-r)-rf f, , will be proportional to will be proportional to the average degree of risk aversion of the the average degree of risk aversion of the investor population and the risk of the market investor population and the risk of the market portfolio. Now we can explain this result.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9-25 在在简简化化了了的的CAPMCAPM模模型型经经济济中中，，无无风风险险投投资资包包括括投投资资者者之之间间的的借借入入与与贷贷出出。

任任何何借借 入入头头寸寸必必须须同同时时有有债债权权人人的的贷贷出出头头寸寸作作为为抵抵偿偿这这意意味味着着投投资资者者之之间间的的净净借借入入与与净净贷贷 出出的的总总和和为为零零那那么么在在风风险险资资产产组组合合上上的的投投资资比比例例总总的的来来说说是是 100%100%，，或或 y y = = 1 1设设y y＝＝1 1，， 代代入入9-19-1式式经经整整理理，，我我们们发发现现市市场场资资产产组组合合的的风风险险溢价与风险厌恶的平均水平有关：溢价与风险厌恶的平均水平有关： In the simplified CAPM economy, risk-free investments involve borrowing and In the simplified CAPM economy, risk-free investments involve borrowing and lending among investors. Any borrowing position must be offset by the lending among investors. Any borrowing position must be offset by the lending position of the creditor. This means that net borrowing and lending lending position of the creditor. This means that net borrowing and lending across all investors must be zero, and in consequence the average position across all investors must be zero, and in consequence the average position in the risky portfolio is 100%, or in the risky portfolio is 100%, or y=y=1. Setting 1. Setting y=y=1 in equation 9.1 and 1 in equation 9.1 and rearranging, we find that the risk premium on the market portfolio is related rearranging, we find that the risk premium on the market portfolio is related to its variance by the average degree of risk aversion: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9-26单个证券的风险益价是单个证券对市场组合风险单个证券的风险益价是单个证券对市场组合风险单个证券的风险益价是单个证券对市场组合风险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的贡献函数的贡献函数的贡献函数 The risk premium on individual securities is a The risk premium on individual securities is a function of the individual securityfunction of the individual security’ ’s contribution s contribution to the risk of the market portfolio.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单个证券的风险益价是构成市场组合资产收益协单个证券的风险益价是构成市场组合资产收益协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方差的函数方差的函数方差的函数 An individual securityAn individual security’ ’s risk premium is a s risk premium is a function of the covariance of returns with the function of the covariance of returns with the assets that make up the market portfolio.assets that make up the market portfolio.单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities9-27 假定现在我们要测算通用公司股票的资产组合风险，我们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（用通用公司股票（GMGM股）股） 同市场资产组合的协方差来刻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，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，先要再次阐明市场资产组合的方差是如何计算的为此，我们按第我们按第 8 8章讨论过的方法将章讨论过的方法将 n n阶协方差矩阵各项按照从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 Suppose, for example, that we want to gauge the portfolio risk of GM stock. We mea- sure the contribution portfolio risk of GM stock. We mea- sure the contribution to the risk of the overall portfolio from holding GM stock to the risk of the overall portfolio from holding GM stock by its covariance with the market portfolio. To see why by its covariance with the market portfolio. To see why this is so, let us look again at the way the variance of the this is so, let us look again at the way the variance of the market portfolio is calculated. To calculate the variance market portfolio is calculated. To calculate the variance of the market port- folio, we use the bordered covariance of the market port- folio, we use the bordered covariance matrix with the market portfolio weights, as discussed in matrix with the market portfolio weights, as discussed in Chapter 8. We highlight GM in this depiction of the Chapter 8. We highlight GM in this depiction of the n n stocks in the market portfolio.stocks in the market portfolio. 单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities9-28单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities9-29单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected 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 contribution of GM’s stock to the variance of the market portfolio is: 9-30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 asThe 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 SecuritiesExpected Return and Risk on Individual Securities9-31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 is: 单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities9-32 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产组合的风险溢价为E(rE(rMM -r -rf f ) )，方差为，方差为σ σ2 2MM ，酬报与波动性比率，酬报与波动性比率为为: : Having measured the contribution of GM stock to market Having measured the contribution of GM stock to market variance, we may determine the appropriate risk premium for variance, we may determine the appropriate risk premium for GM. We note first that the market portfolio has a risk premium GM. We note first that the market portfolio has a risk premium of E(rof E(rMM -r -rf f ) ) and a variance of and a variance of σ σ2 2MM , for a reward-to-risk ratio of , for a reward-to-risk ratio of单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities9-33 假定某位平均的投资者投资于市场资产组合的比例为假定某位平均的投资者投资于市场资产组合的比例为 100%100%，现在他打算通过借入无风险贷款的方式来增加，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。

新的资产组合由以比例为小量的市场资产组合头寸新的资产组合由以下三下三 部分组成：收益为部分组成：收益为 r rMM的原有市场资产组合头寸，的原有市场资产组合头寸，收益为收益为 - -δ δr rf f 的无风险资产空头头寸，以及收益为的无风险资产空头头寸，以及收益为δ δr rMM的的市场资产组合的多头头寸总的资产组合收益为市场资产组合的多头头寸总的资产组合收益为r rMM＋＋ δ δ( (r rMM- -r rf f) )，将其期望值与最初期望值，将其期望值与最初期望值 E E( (r rMM) )比较，期望收比较，期望收益的增加额为益的增加额为单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities9-34单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities9-35单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected 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的方差新资产组合由权重为(1(1＋＋δ δ) )的市场资产组合的市场资产组合与权重为与权重为- - δ δ的无风险资产组成，调整后的资产组合的的无风险资产组成，调整后的资产组合的方差为方差为: : To measure the impact of the portfolio shift on risk, To measure the impact of the portfolio shift on risk, we compute the new value of the portfolio variance. we compute the new value of the portfolio variance. The new portfolio has a weight of (1+The new portfolio has a weight of (1+δ δ) in the market ) in the market and -and -δ δ in the risk-free asset. Therefore, the variance in the risk-free asset. Therefore, the variance of the adjusted portfolio is:of the adjusted portfolio is:9-36单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities 由于δ非常小，所以相比于2δ而言δ2可以忽略，因而我们这里对这一项忽略不计 However, if δ is very small, then δ2 will be negligible compared to 2δ , so we may ignore this term 9-37单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected 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, referred to as the marginal price of risk, is given by the ratio 9-38单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected 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的资金 投资于通用公司股票。

他的平均超额收益的增投资于通用公司股票他的平均超额收益的增加值为加值为: : Now suppose that, instead, investors were to invest the Now suppose that, instead, investors were to invest the incrementincrementin GM stock, also financed by in GM stock, also financed by borrowing at the risk-free rate. The increase in mean borrowing at the risk-free rate. The increase in mean excess return is:excess return is:9-39单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected 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δ ，投资于无风险资产的资金权重为-δ。

这一资产组合的方差为: This portfolio has a weight of 1.0 in the market, δ in GM, and -δ in the risk-free asset. Its GM variance is: 9-40单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and 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: 9-41单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities δ2忽略不计，通用公司股票的风险边际价格为 Dropping the negligible term involving δ2, the marginal price of risk of GM is9-42单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected 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：险边际价格相等的等式如下： Equating the marginal price of risk of GM’s stock to Equating the marginal price of risk of GM’s stock to that of the market results in a relationship between the that of the market results in a relationship between the risk premium of GM and that of the market:risk premium of GM and that of the market:9-43单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected 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9-44单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities 这里，这里，Cov(Cov(r rGMGM，，r rMM)/ )/ σ σ2 2MM测度的是通用公司股票对市场资产组合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。

这一方差的贡献程度，这是市场资产组合方差的一个组成部分这一比率称作贝塔（比率称作贝塔（betabeta），以），以β β表示，这样，表示，这样，9-69-6式可以写作为：式可以写作为： The ratio Cov( The ratio Cov(r rGMGM，，r rMM)/ )/ σ σ2 2MM measures the contribution of GM measures the contribution of GM stock to the variance of the market portfolio as a fraction of the stock to the variance of the market portfolio as a fraction of the total variance of the market portfolio. The ratio is called total variance of the market portfolio. The ratio is called beta beta and is denoted by and is denoted by β β. Using this measure, we can restate . Using this measure, we can restate equation 9.6 asequation 9.6 as9-45单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities 上式即是上式即是 CAPMCAPM模型的最普通形式模型的最普通形式─ ─ 期望收益期望收益- -贝塔关贝塔关系系, , 我们对这一关系式还要做更详尽的论述。

我们对这一关系式还要做更详尽的论述 This This expected returnexpected return– –beta relationship beta relationship is the most is the most familiar expression of the CAPM to practitioners. We familiar expression of the CAPM to practitioners. We will have a lot more to say about the expected return–will have a lot more to say about the expected return–beta relationship shortly.beta relationship shortly.9-46单个证券的收益和风险Expected Return and Risk on Individual SecuritiesExpected Return and Risk on Individual Securities If the expected return–beta relationship holds for any individual If the expected return–beta relationship holds for any individual asset, it must hold for any combination of assets. Suppose that asset, it must hold for any combination of assets. Suppose that some portfolio some portfolio P P has weight has weight wk wk for stock for stock k, k, where where k k takes on takes on values 1, . . . , values 1, . . . , n n. Writing out the CAPM equation 9.7 for each . Writing out the CAPM equation 9.7 for each stock, and multiplying each equation by the weight of the stock stock, and multiplying each equation by the weight of the stock in the portfolio, we obtain these equations, one for each stock:in the portfolio, we obtain these equations, one for each stock:9-47证券市场线证券市场线Security Market LineE(r)E(rM)rf证券市场线证券市场线SMLb bM= 1.09-48= [COV(ri,rm)] / m2证券市场线斜率证券市场线斜率证券市场线斜率证券市场线斜率Slope SML =E(rm) - rf =市场风险溢价市场风险溢价market risk premium SML = rf + [E(rm) - rf] Betam = [Cov (ri,rm)] / m2 = m2 / m2 = 1:证券的协方差风险证券市场线关系证券市场线关系SML Relationships9-49β β系数。

美国经济学家威廉系数美国经济学家威廉· ·夏普提出的风险衡量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）作为测量股票指（市场组合）作为测量股票β β值的基准）值的基准）如果如果β β值为值为1.11.1，即表明该股票波动性要比市场大，即表明该股票波动性要比市场大盘高盘高1010％，说明该股票的风险大于市场整体的风％，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进攻型证券反之则是防守型股票无风险证券的的β β值等于零，市场组合相对于自身的值等于零，市场组合相对于自身的β β值为值为1 19-50E(r E(r i i ) = r ) = rf f + +   i i(E(r(E(rmm) - r) - rf f ) )资本资产定价模型的最普通形式资本资产定价模型的最普通形式资本资产定价模型的最普通形式资本资产定价模型的最普通形式——期望收益贝塔关系期望收益贝塔关系期望收益贝塔关系期望收益贝塔关系E(rE(rmm) - r) - rf f = .08 = .08 r rf f = .03= .03 x x = 1.25= 1.25E(rE(rx x) = .03 + 1.25(.08) = .13 or 13%) = .03 + 1.25(.08) = .13 or 13% y y = .6 = .6E(rE(ry y) = .03 + .6(.08) = .078 or 7.8%) = .03 + .6(.08) = .078 or 7.8%证券市场线计算实例证券市场线计算实例Sample Calculations for SML9-51计算图形Graph of Sample CalculationsE(r)Rx=13%SMLb b1.0Rm=11%Ry=7.8%3%1.25b bx.6b by.08»By9-52布莱克的零贝塔模型Black’s Zero Beta Model缺少无风险资产缺少无风险资产 Absence of a risk-free assetAbsence of a risk-free asset在有效边界上的任何资产组合是有效资产组合。

在有效边界上的任何资产组合是有效资产组合 Combinations of portfolios on the efficient frontier are Combinations of portfolios on the efficient frontier are efficient. efficient. 有效率边界上的任一资产组合有不相关组合相伴有效率边界上的任一资产组合有不相关组合相伴 All frontier portfolios have companion portfolios that All frontier portfolios have companion portfolios that are uncorrelated. are uncorrelated. 任何单个资产的收益可以准确地由任意两个边界资产任何单个资产的收益可以准确地由任意两个边界资产组合的期望收益的线性函数表示组合的期望收益的线性函数表示 任何单个资产的收益可以由有效组合的线性函数表示任何单个资产的收益可以由有效组合的线性函数表示 Returns on individual assets can be expressed as Returns on individual assets can be expressed as linear combinations of efficient portfolios.linear combinations of efficient portfolios. 9-53布莱克的零贝塔模型Black’s Zero Beta Model 布莱克的禁止卖空无风险资产的布莱克的禁止卖空无风险资产的 CAPMCAPM模型建立在下列三项有效率资产组合模型建立在下列三项有效率资产组合的方的方 差均值性质之上：差均值性质之上： 1) 1) 任何有效率资产组合组成的资产组合仍然是有效率资产组合。

任何有效率资产组合组成的资产组合仍然是有效率资产组合 2) 2)有效率边界上的任一资产组合在最小方差边界的下半部分（无效率部分）有效率边界上的任一资产组合在最小方差边界的下半部分（无效率部分）上均上均 有相应的有相应的“ “伴随伴随” ”资产组合存在，由于这些资产组合存在，由于这些“ “伴随伴随” ”资产组合是不相资产组合是不相关的，因此，这关的，因此，这 些资产组合可以被视为有效率资产组合中的零贝塔资产组些资产组合可以被视为有效率资产组合中的零贝塔资产组合 Black’s model of the CAPM in the absence of a risk-free asset rests on the Black’s model of the CAPM in the absence of a risk-free asset rests on the three following properties of mean-variance efficient portfolios:three following properties of mean-variance efficient portfolios: 1. Any portfolio constructed by combining efficient portfolios is itself on the 1. Any portfolio constructed by combining efficient portfolios is itself on the efficient frontier.efficient frontier. 2. Every portfolio on the efficient frontier has a “companion” portfolio on 2. Every portfolio on the efficient frontier has a “companion” portfolio on the bottom half (the inefficient part) of the minimum-variance frontier with the bottom half (the inefficient part) of the minimum-variance frontier with which it is uncorrelated. Because the portfolios are uncorrelated, the which it is uncorrelated. Because the portfolios are uncorrelated, the companion portfolio is referred to as the companion portfolio is referred to as the zero-beta portfolio zero-beta portfolio of the of the efficient portfolio.efficient portfolio.9-54有效组合和零贝塔伴随Efficient Portfolios and Zero CompanionsQPZ(Q)Z(P)E[rz (Q)]E[rz (P)]E(r) 9-55布莱克的零贝塔模型方程Black’s Zero Beta Model Formulation任何资产的期望收益可以准确地由任意两个边界资产组合的期望收益的线性函 数表示。

例如，考虑有两个最小方差边界资产组合 P与Q，布莱克给出任意资产i的期 望收益的表达如下： The expected return of any asset can be expressed as an exact, linear function of the expected return on any two frontier portfolios. Consider, for example, the minimum-variance frontier portfolios P and Q. Black showed that the expected return on any asset i can be expressed as9-56布莱克的零贝塔模型方程Black’s Zero Beta Model Formulation假定经济中只有两个投资者，一个相对来说厌恶风险，而另外一个可以忍受风险 厌恶风险的投资者选择资本配置线上的资产组合 T，如图 9-5所示，也就是说，他的资 产组合由资产组合 T与按无风险利率贷出的无风险资产组成 T是由无风险借贷利率 rf 出发的有效率边界的切点。

忍受风险的投资者愿意在承担更多风险的前提下取得更高 的风险溢价：他选择图中的 SS资产组合与 T资产组合相比较，虽同处于有效率边界 但其风险与收益均高于 T 资产组合总的风险资产组合（也就是市场资产组合，M ）由T与S结合而成，各自权重由两个投资者的相对财富与风险厌恶程度决定由于 T与S 都在有效率边界上，所以根据性质 1，市场资产组合 M也在有效率边界上9-57布莱克的零贝塔模型方程Black’s Zero Beta Model FormulationImagine an economy with only two investors, one relatively risk averse and one risk tolerant. The risk-averse investor will choose a portfolio on the CAL supported by portfolio T in Figure 9.8, that is, he will mix portfolio T with lending at the risk-free rate. T is the tangency portfolio on the efficient frontier from the risk-free lending rate, rf. The risk-tolerant investor is willing to accept more risk to earn a higher-risk premium; she will choose portfolio S. This portfolio lies along the efficient frontier with higher risk and return than portfolio T. The aggregate risky portfolio (i.e., the market portfolio, M) will be a combination of T and S, with weights determined by the relative wealth and degrees of risk aversion of the two investors. Since T and S are each on the efficient frontier, so is M (from Property 1).9-58布莱克的零贝塔模型方程Black’s Zero Beta Model Formulation9-59布莱克的零贝塔模型方程Black’s Zero Beta Model Formulation根据性质 2，市场资产组合 M 也存在一个在最小方差边界上的零贝塔“伴随”资 产组合： Z(M)，见图 9-5。

根据性质 3及9-8式，我们可以用市场资产组合 M及Z(M)来表 示任何证券的收益由于 Cov(r M，r Z(M) )＝0，所以有From Property 2, M has a companion zero-beta portfolio on the minimum-variance frontier, Z(M), shown in Figure 9.8. Moreover, by Property 3 we can express the return on any security in terms of M and Z(M) as in equation 9.8. But, since by construction Cov(r M，r Z(M) )＝0 , the expression simplifies to9-60零贝塔市场模型Zero Beta Market Model式中的资产组合 P与资产组合 Q分别由市场资产组合 M及Z(M)代替上式可视为一个简化了的 CAPM模型，在其中，E(r z (m)) 取代了rf where P has been replaced by M and Q has been replaced by Z(M). Equation 9.9 may be interpreted as a variant of the simple CAPM, in which r f has been replaced with E(r z (m)) 9-61资本资产定价模型和流动性CAPM & Liquidity流动性流动性 流动性是指资产转化为现金时所需的费用与便捷流动性是指资产转化为现金时所需的费用与便捷程度。

交易者非常注重流动性，一些研究证实缺程度交易者非常注重流动性，一些研究证实缺乏流动性将大大降低资产的市场出售价格水平乏流动性将大大降低资产的市场出售价格水平 Liquidity Liquidity Liquidity refers to the cost and ease with which Liquidity refers to the cost and ease with which an asset can be converted into cash, that is, an asset can be converted into cash, that is, sold. Traders have long recognized the sold. Traders have long recognized the importance of liquidity, and some evidence importance of liquidity, and some evidence suggests that illiquidity can reduce market suggests that illiquidity can reduce market prices substantially. prices substantially. 9-62资本资产定价模型和流动性CAPM & Liquidity非流动溢价非流动溢价Illiquidity Premium Illiquidity Premium 流动性差的资产低价交易，流动性高的资产期望流动性差的资产低价交易，流动性高的资产期望收益也高，流动性效用的大小同资产的交易费用收益也高，流动性效用的大小同资产的交易费用分布状况以及投资者投资内容的分布有关。

分布状况以及投资者投资内容的分布有关 illiquid assets trade at lower prices or, illiquid assets trade at lower prices or, equivalently, that the expected return on illiquid equivalently, that the expected return on illiquid assets must be higher. assets must be higher. 研究支持非流动溢价研究支持非流动溢价 Research supports a premium for illiquidity. Research supports a premium for illiquidity.– –Amihud and MendelsonAmihud and Mendelson9-63流动溢价的资本资产定价模型CAPM with a Liquidity PremiumE (ri ) – rf = βi [ E(rM ) - r f ] + f(ci)f (ci) = 证券i 的流动溢价， f (ci) 是关于ci的一阶单调递增函数。

liquidity premium for security i9-64非流动性与平均收益关系Illiquidity and Average Returns平均月收益率平均月收益率Average monthly return(%)买卖差价买卖差价Bid-ask spread (%)9-65Summary CAPM 模型假定所有投资者均为单期投资，并且遵循相同的投资构，并力求获得具有最小方差的最优资产组合 The CAPM assumes that investors are single-period planners who agree on a common input list from security analysis and seek mean-variance optimal portfolios.9-66Summary CAPM CAPM模型假定理想状态下的股票市场具有以下特征：模型假定理想状态下的股票市场具有以下特征： a. a. 股票市场容量足够大，并且其中所有的投资者为价格股票市场容量足够大，并且其中所有的投资者为价格接受者。

接受者 b. b. 不存在税收与交易费用不存在税收与交易费用 c. c. 所有风险资产均可公开交易所有风险资产均可公开交易 d. d. 投资者可以以无风险利率借入或贷出任意额度资产投资者可以以无风险利率借入或贷出任意额度资产 The CAPM assumes that security markets are ideal in The CAPM assumes that security markets are ideal in the sense that:the sense that: a. They are large, and investors are price-takers. a. They are large, and investors are price-takers. b. There are no taxes or transaction costs. b. There are no taxes or transaction costs. c. All risky assets are publicly traded. c. All risky assets are publicly traded. d. Investors can borrow and lend any amount at a fixed d. Investors can borrow and lend any amount at a fixed risk-free rate.risk-free rate.9-67SUMMARY 根据以上假定，投资者持有无差异的风险资产组合。

CAPM模型认为市场资产组合是唯一的具有最小方差的有相切的资产组合，所以消极的投资策略是有效的 With these assumptions, all investors hold identical risky portfolios. The CAPM holds that in equilibrium the market portfolio is the unique mean-variance efficient tangency portfolio. Thus a passive strategy is efficient.9-68SUMMARY CAPM 模型中的市场资产组合是市值加权资产组合，其意义为所有股票在资产组合中的权重等于该股票的流通市值占总市值的比重 The CAPM market portfolio is a value-weighted portfolio. Each security is held in a proportion equal to its market value divided by the total market value of all securities.9-69SUMMARY 如果市场资产组合有效且投资者平均无借入或贷出如果市场资产组合有效且投资者平均无借入或贷出行为，则市场资产组合的风险溢价正比于其方差行为，则市场资产组合的风险溢价正比于其方差σ σMM2 2，投资者风险厌恶的平均相关系数，投资者风险厌恶的平均相关系数 A A：： If the market portfolio is efficient and the average If the market portfolio is efficient and the average investor neither borrows nor lends, then the risk investor neither borrows nor lends, then the risk premium on the market portfolio is proportional to premium on the market portfolio is proportional to its variance, its variance, σ σMM2 2 , and to the average coefficient of , and to the average coefficient of risk aversion across investors, risk aversion across investors, A A: : E E ( (r r M M ) – ) – r r f f = 0.01 = 0.01 A A σ σMM2 2 9-70SUMMARY CAPM CAPM 模型认为任意单个资产或资产组合的风险溢价为市场资产组合的模型认为任意单个资产或资产组合的风险溢价为市场资产组合的风险溢风险溢 价与贝塔系数的乘积：价与贝塔系数的乘积： The CAPM implies that the risk premium on any individual asset or The CAPM implies that the risk premium on any individual asset or portfolio is the product of the risk premium on the market portfolio and portfolio is the product of the risk premium on the market portfolio and the beta coefficient:the beta coefficient: E (r i ) – r f = βi [ E(r M ) - r f ] 这里，贝塔系数等于作为市场资产组合方差一部分的单个资产同市场资这里，贝塔系数等于作为市场资产组合方差一部分的单个资产同市场资产组合的协方差：产组合的协方差： where the beta coefficient is the covariance of the where the beta coefficient is the covariance of the asset with the market portfolio as a fraction of the asset with the market portfolio as a fraction of the variance of the market portfoliovariance of the market portfolio β βi i= =Cov( Cov( r ri i , , r rMM )/ )/σ σMM2 29-71SUMMARY 在在 CAPMCAPM模型其他假定不变的条件下，当无风险模型其他假定不变的条件下，当无风险资产借入或贷出受限制时，资产借入或贷出受限制时， CAPMCAPM模型的简单形式模型的简单形式修正为零贝塔修正为零贝塔 CAPMCAPM模型。

零贝塔资产组合期望模型零贝塔资产组合期望收益率取代期收益率取代期 望收益望收益 - -贝塔关系中的无风险利率：贝塔关系中的无风险利率： When risk-free investments are restricted but all When risk-free investments are restricted but all other CAPM assumptions hold, then the simple other CAPM assumptions hold, then the simple version of the CAPM is replaced by its zero-beta version of the CAPM is replaced by its zero-beta version. Accordingly, the risk-free rate in the version. Accordingly, the risk-free rate in the expected return–beta relationship is replaced by expected return–beta relationship is replaced by the zero-beta port- folio’s expected rate of return:the zero-beta port- folio’s expected rate of return: E (r i ) = E[ r Z( M) ] + βi E[ r M - r Z( M) ]9-72SUMMARY CAPM CAPM 模型的简单形式假定投资者均是短视的行为人。

当模型的简单形式假定投资者均是短视的行为人当投资者根据生命期及保留遗产来制定个人投资计划时，只投资者根据生命期及保留遗产来制定个人投资计划时，只要投资人的偏好及股票收益率分布不变，市场资产组合就要投资人的偏好及股票收益率分布不变，市场资产组合就仍旧有效，并且仍旧有效，并且CAPMCAPM模型的简单形式及期望收益模型的简单形式及期望收益 - -贝塔关贝塔关系仍然适用系仍然适用 The simple version of the CAPM assumes that investors The simple version of the CAPM assumes that investors are myopic. When investors are assumed to be concerned are myopic. When investors are assumed to be concerned with lifetime consumption and bequest plans, but with lifetime consumption and bequest plans, but investors’ tastes and security return distributions are investors’ tastes and security return distributions are stable over time, the market portfolio remains efficient and stable over time, the market portfolio remains efficient and the simple version of the expected return–beta the simple version of the expected return–beta relationship holds.relationship holds.9-73SUMMARY 流动费用可以被吸收进流动费用可以被吸收进 CAPMCAPM模型。

在存在大量具模型在存在大量具有贝塔与流动费用有贝塔与流动费用 c c i i 任意组任意组 合的资产的情况下，合的资产的情况下，期望收益根据下式会哄抬以反映这一非意愿的性质：期望收益根据下式会哄抬以反映这一非意愿的性质： Liquidity costs can be incorporated into the Liquidity costs can be incorporated into the CAPM relationship. When there is a large number CAPM relationship. When there is a large number of assets with any combination of beta and of assets with any combination of beta and liquidity cost liquidity cost c c i i, the expected return is bid up to , the expected return is bid up to reflect this undesired property according toreflect this undesired property according to E (r i ) – r f = βi [ E(rM ) - r f ] + f ( c i )。