多因素模型与套利定价理论

1、1 多因素模型2 套利定价理论1. 多因素模型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）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。iiiir = E(r )+F+e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。多因素模型ri = E(ri) + iGDP GDP + iIR IR + ei ri = Return for security i iGDP= Factor sen

2、sitivity for GDP iIR = Factor sensitivity for Interest Rate ei = Firm specific eventsiiiGDPiIRir = E(r )+bGDP+b IR+e电力公司 公共事业对GDP的值较小，而对于利率却有较高的敏感度。航空公司 对经济活动敏感，对利率不敏感。消息表明经济将持续扩张，预期GDP和利率都将上升，对于两个公司的影响？r=0.1+1.8GDP+0.7IR+e南方航空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的值）乘以GDP风险的风险溢价对利率风险的敏感度（利率的值）乘以利率风险的风险溢价i

3、fGDPGDPIRIRGDPGDPfr = r +RP+RPRP= r-r2. 套利定价理论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。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

4、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。APT 充分分散化的资产组合充分分散化的资产组合rP = E (rP) + PF + ePF = some factorFor a well-diversified portfolio: eP approaches zeroSimilar to CAPM,Figure 10.1 Returns as a Function of the Systematic FactorFigure 10.2 值相等值相等Figure 10.3 值不相等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

5、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CAPMAPT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。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 Line4. 多因素套利定价理论多因素套利定价理论多因素资本资产定价模型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

6、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）Two-Factor ModelThe multifactor APR is similar to the one-factor case But need to think in terms of a factor portfolioWell-diversifiedBeta of 1 for one factorBeta of 0 for any other1122( )iiiiirE rFFeMultifactor Model Equationri = E(ri) + iGDP GDP + iIR IR + ei ri =

7、 Return for security i iGDP= Factor sensitivity for GDP iIR = Factor sensitivity for Interest Rate ei = Firm specific eventsMultifactor SML ModelsE(r) = rf + GDPRPGDP + IRRPIR GDP = Factor sensitivity for GDP RPGDP = Risk premium for GDP IR = Factor sensitivity for Interest RateRPIR = Risk premium f

8、or Interest RateiiiiArbitrage Pricing TheoryArbitrage - arises if an investor can construct a zero investment portfolio with a sure profitSince no investment is required, an investor can create large positions to secure large levels of profitIn efficient markets, profitable arbitrage opportunities w

9、ill quickly disappearAPT & Well-Diversified PortfoliosrP = E (rP) + PF + ePF = some factorFor a well-diversified portfolio: eP approaches zeroSimilar to CAPM,APT applies to well diversified portfolios and not necessarily to individual stocksWith APT it is possible for some individual stocks to b

10、e mispriced - not lie on the SMLAPT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolioAPT can be extended to multifactor modelsAPT and CAPM ComparedMultifactor APTUse of more than a single factorRequires formation of factor port

11、foliosWhat factors?Factors that are important to performance of the general economyFama-French Three Factor ModelTwo-Factor ModelThe multifactor APR is similar to the one-factor case But need to think in terms of a factor portfolioWell-diversifiedBeta of 1 for one factorBeta of 0 for any other1122(

12、)iiiiirE rFFeExample of the Multifactor ApproachWork of Chen, Roll, and RossChose a set of factors based on the ability of the factors to paint a broad picture of the macro-economyAnother Example:Fama-French Three-Factor ModelThe factors chosen are variables that on past evidence seem to predict ave

13、rage returns well and may capture the risk premiumsWhere:SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocksHML = 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 ratioitiiMMtiSMBtiHMLtitrRSMBHMLeThe Multifactor CAPM a