多因素模型與套利定價理論

1、1 多因素模型2 套利定價理論1. 多因素模型多因素模型市場證券組合收益概括了宏觀因素的重要影響。（單因素模型）單因素模型認(rèn)為每一種股票對每種風(fēng)險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時期影響證券收益率的因素。多因素模型允許每個股票對于不同的宏觀因素的具有不同的敏感度，即不同的值。多因素模型還可以應(yīng)用于風(fēng)險管理。它提供了一個衡量宏觀經(jīng)濟(jì)風(fēng)險的簡單方法，并且構(gòu)造證券組合來規(guī)避那些風(fēng)險。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電力公司 公共事業(yè)對GDP的值較小，而對于利率卻有較高的敏感度。航空公司 對經(jīng)濟(jì)活動敏感，對利率不敏感。消息表明經(jīng)濟(jì)將持續(xù)擴(kuò)張，預(yù)期GDP和利率都將上升，對于兩個公司的影響？r=0.1+1.8GDP+0.7IR+e南方航空證券的收益率可以分為：無風(fēng)險收益率對GDP風(fēng)險的敏感度（GDP的值）乘以GDP風(fēng)險的風(fēng)險溢價對利率風(fēng)險的敏感度（利率的值）乘以利率風(fēng)險的風(fēng)險溢價i

3、fGDPGDPIRIRGDPGDPfr = r +RP+RPRP= r-r2. 套利定價理論套利定價理論基本假設(shè)證券收益可以用單因素模型表示市場上有足夠多的證券來分散不同的風(fēng)險功能完善的證券市場消滅持續(xù)的套利機(jī)會套利當(dāng)投資者可以得到無風(fēng)險利潤，而不必做凈投資時，就出現(xiàn)了套利機(jī)會。無風(fēng)險套利資產(chǎn)組合的重要性質(zhì)：任何投資者不考慮風(fēng)險厭惡或財(cái)富狀況，都愿意盡可能地?fù)碛性撡Y產(chǎn)組合的頭寸。市場價格會變動至套利機(jī)會消除。證券價格應(yīng)該滿足“無套利”條件，即要滿足不存在套利機(jī)會的價格水平。在特定領(lǐng)域比如并購目標(biāo)股票的搜尋中，尋找定價出現(xiàn)偏差的證券的專業(yè)行為。（風(fēng)險套利）衍生證券市場價值完全由其他證券的價格來決

4、定，因此，無套利條件可以導(dǎo)致準(zhǔn)確的定價。股票不是由其他資產(chǎn)的價格決定的，無套利條件須從分散化投資中導(dǎo)出。APT 充分分散化的資產(chǎn)組合充分分散化的資產(chǎn)組合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 值不相等值不相等當(dāng)所有充分分散投資組合的期望收益率位于圖中通過無

5、風(fēng)險資產(chǎn)點(diǎn)的直線上。這條直線的方程給出了所有充分分散化投資組合的期望收益值。APT與CAPMAPT不要求證券市場線關(guān)系的基準(zhǔn)資產(chǎn)組合是真實(shí)市場的投資組合。APT為證券市場線關(guān)系的實(shí)際實(shí)現(xiàn)中利用指數(shù)模型提供了進(jìn)一步理由。（只要指數(shù)組合是充分分散化的，證券市場線關(guān)系仍然可以真實(shí)地與APT保持一致。單項(xiàng)資產(chǎn)與APT如果所有充分分散化的投資組合均滿足該關(guān)系，那么所有的單個證券也將幾乎肯定地滿足這個關(guān)系。Figure 10.4 The Security Market Line4. 多因素套利定價理論多因素套利定價理論多因素資本資產(chǎn)定價模型多因素資本資產(chǎn)定價模型因素的來源勞動收入的不確定性重要消費(fèi)品價格的

6、不確定性（如能源價格）未來投資機(jī)會的變化（如各種資產(chǎn)風(fēng)險等級的變化）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