Finance is not my expertise. I only know three basic models for asset pricing: Capital asset pricing model (CAPM), Arbitrage Pricing Theory (APT), and Fama and French (FF) model. CAPM argues that investor should be compensated in two ways: time value of money and risk premium. In specific, the require rate of return of a certain asset should be: Ri=Rf+Beta(Rm-Rf) Where, Ri is the required rate of return of a certain asset. Rf means risk free rate of return, such as the rate of return of government bonds. Rm is the average market return rate. Beta measures the sensitivity of an asset return on market returns. CAPM showed that portfolio returns can be optimized by investing adverse related assets, where optimize means to have same rate of return in lowest risk level. There are many critiques on CAPM. The most famous one is from Roll. Roll’s critique argues that investors tend invest more in home market rather than foreign market, called home bias. Thus, beta could be little use when consider international portfolio. Also, CAPM has many assumptions which make it hard to apply for real business. Assumptions of CAPM:1)Frictionless markets. 2)All information is easily available. 3)No transaction costs. 4)No taxes. 5)All investors can borrow and lend at the risk free rate. 6)Investors have homogeneous expectations regarding future expected returns and risk. 7)Rational investors

Ross proposed arbitrage pricing theory (APT). He stated that required rate of return should not be determined by only Rf and Rm. There should be more factors. Those factors should be specific for that asset. APT model can be expressed as: Ri=Rf+beta1*Factor1+beta2*Factor 2+…+betaN*factorN. APT did not show what exacted those factors are. It depends on which asset you measure. For example, the factors of EXXON Mobile will not be the same as Bank of America. Due to the Ambiguity of APT, it is not very popular as CAPM.

As an extension of CAPM, Fama and French add book to market ratio and size into consideration. Many previous researches showed that value stock outperforms growth stock; and small cap stock outperforms large cap stock. This phenomenon is observed worldwide. Thus, in their Fama and French model, Ri=Rf+Beta1*(Rm-Rf)+Beta2*SML+Beta3*HML, where SML means small minus large, HML means high book to market minus low book to market.

Avramov and Chordia (2006) tested different types of asset pricing models on book to market(BM), size, turnover, and past returns. Their results showed that CAPM, CCAPM fail to capture all BM, size, turnover, and past returns. Fama and French with constant risk factor also fail to capture those effects. But conditional Fama and French model did capture BM and Size effects. Ng (2004) created Dynamic international CAPM based on Dynamic CAPM, International CAPM, and basic CAPM. Avramov (2004) add investor prior belief into consideration. He classifies investor prior belief into: skeptical, mutual, and confidential. FF model and its extension version are used in this research. His results showed that conditional FF outperform others, while mutual view outperform skeptical and confidential. Aretz (2010) examine book to market and size with macroeconomic factors. They found that book to market captures the growth expectation and term structure. Size captured default risk. Archarya and Padrsen (2005) add liquidity into CAPM and created liquidity adjusted CAPM. Their empirical test showed their model performs better than CAPM.

I am not a finance guy. But stock investment can be regarded as the result of people's investing behavior. In this sense, we have to admit that stock market is a dynamic system. Using linear functions to predict dynamic system is like using arithmetic to do fortune-telling. But the interesting thing is that the linear models themselves are also in this dynamic system and have impact on investor's behavior. When people believe in those models, they do works. Ex. If everyone believes the movement of sunspot can predict stock price, we will find significant correlation between them.