APT & Well-Diversified Portfolios
rP = E (rP) + βPF + eP F = some factor For a well-diversified portfolio: eP approaches zero Similar to CAPM,
APT and CAPM Compared
多因素模型
ri = E(ri )+biGDP GDP +biIR IR + ei
ri = E(ri) + βiGDP GDP + βiIR IR + ei = Return for security i βiGDP= Factor sensitivity for GDP βiIR = Factor sensitivity for Interest Rate ei = Firm specific events ri
Two-Factor Model
ri = E(ri ) + βi1F + βi2F2 + ei 1
The multifactor APR is similar to the onefactor case
But need to think in terms of a factor portfolio
市场价格会变动至套利机会消除。证券价 格应该满足“无套利”条件，即要满足不 存在套利机会的价格水平。 在特定领域比如并购目标股票的搜寻中， 寻找定价出现偏差的证券的专业行为。 （风险套利）
衍生证券 市场价值完全由其他证券的价格来决定， 因此，无套利条件可以导致准确的定价。 股票 不是由其他资产的价格决定的，无套利条 件须从分散化投资中导出。
Two-Factor Model
ri = E(ri ) + βi1F + βi2F2 + ei 1
The multifactor APR is similar to the onefactor case
But need to think in terms of a factor portfolio
Well-diversified Beta of 1 for one factor Beta of 0 for any other
Multifactor Model Equation
ri = E(ri) + βiGDP GDP + βiIR IR + ei = Return for security i βiGDP= Factor sensitivity for GDP βiIR = Factor sensitivity for Interest Rate ei = Firm specific events ri
Figure 10.2 β值相等
Figure 10.3 β值不相等
当所有充分分散投资组合的期望收益率位 于图中通过无风险资产点的直线上。这条 直线的方程给出了所有充分分散化投资组 合的期望收益值。
APT与CAPM APT不要求证券市场线关系的基准资产组合 是真实市场的投资组合。 APT为证券市场线关系的实际实现中利用指 数模型提供了进一步理由。（只要指数组 合是充分分散化的，证券市场线关系仍然 可以真实地与APT保持一致。
Well-diversified Beta of 1 for one factor Beta of 0 for any other
Example of the Multifactor Approach
Work of Chen, Roll, and Ross Chose a set of factors based on the ability of the factors to paint a broad picture of the macro-economy
单项资产与APT 如果所有充分分散化的投资组合均满足该 关系，那么所有的单个证券也将几乎肯定 地满足这个关系。
Figure 10.4 The Security Market Line
4. 多因素套利定价理论
多因素资本资产定价模型
因素的来源
劳动收入的不确定性 重要消费品价格的不确定性（如能源价格） 未来投资机会的变化（如各种资产风险等级的 变化）
Another Example: Fama-French Three-Factor Model
The factors chosen are variables that on past evidence seem to predict average returns well and may capture the risk premiums
Multifactor APT
Use of more than a single factor Requires formation of factor portfolios What factors? Factors that are important to performance of the general economy Fama-French Three Factor Model
Where: SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks HML = High Minus Low, i.e., the return of a portfolio of stocks with a high book to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio
APT applies to well diversified portfolios and not necessarily to individual stocks With APT it is possible for some individual stocks to be mispriced - not lie on the SML APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio APT can be extended to multifactor models
多因素模型和套利定价理论
1 多因素模型 2 套利定价理论
1. 多因素模型
市场证券组合收益概括了宏观因素的重要 影响。（单因素模型） ri = E(ri )+ β i F + ei 单因素模型认为每一种股票对每种风险因 素都有相同的敏感度。
多因素模型可以描述和量化任何时期影响证券收 益率的因素。 多因素模型允许每个股票对于不同的宏观因素的 具有不同的敏感度，即不同的β值。 多因素模型还可以应用于风险管理。它提供了一 个衡量宏观经济风险的简单方法，并且构造证券 组合来规避那些风险。
证券的收益率可以分为：
无风险收益率 对GDP风险的敏感度（GDP的β值）乘以GDP风险的风 险溢价 对利率风险的敏感度（利率的β值）乘以利率风险的风 险溢价
2. 套利定价理论
基本假设
证券收益可以用单因素模型表示 市场上有足够多的证券来分散不同的风险 功能完善的证券市场消灭持续的套利机会
套利 当投资者可以得到无风险利润，而不必做 净投资时，就出现了套利机会。 无风险套利资产组合的重要性质：任何投 资者不考虑风险厌恶或财富状况，都愿意 尽可能地拥有该资产组合的头寸。
rit = αi + βiM RMt + βiSMBSMBt + βiHMLHMLt + eit
The Multifactor CAPM and the APM
A multi-index CAPM will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge The APT is largely silent on where to look for priced sources of risk
Multifactor SML Models E(r) = rf + GDPRPGDP + IRRPIR βi βi
βi = Factor sensitivity for GDP GDP
RPGDP = Risk premium for GDP βi IR = Factor sensitivity for Interest Rate RPIR = Risk premium for Interest Rate
APT 充分分散化的资产组合
rP = E (rP) + βPF + eP F = some factor For a well-diversified portfolio: eP approaches zero Similar to CAPM,
Figure 10.1 Returns as a Function of the Systematic Factor
r南方航空 = 0.1+1.8GDP +0.7IR + e
电力公司 公共事业对GDP的β值较小，而对于 利率却有较高的敏感度。 航空公司 对经济活动敏感，对利率P和利率 都将上升，对于两个公司的影响？
ri = rf + βGDP RPGDP + β IR RPIR RPGDP = rGDP - rf