Single-Factor Model

In a single-factor model, uncertainty in asset returns has two sources: a common or macroeconomic factor, and firm-specific events.

把收益的构成两个部分，一个是可以理解的，一个是不能理解的。

The factor model states that the actual return on firm i will equal its initially expected return plus a (zero expected value) random amount attributable to unanticipated economy-wide events, plus another (zero expected value) random amount attributable to firm–specific events.

单因素模型认为非系统性风险也应该有收益补偿。

$$
R_i=E\left(R_i\right)+\beta_i F+e_i
$$

$F$ is the deviation【偏差】 of the common factor from its expected value;
$\beta_i$ is the sensitivity 【敏感程度/每单位偏差对应的收益率补偿】of firm $i$ to that factor;
$e_i$ is the firm-specific disturbance【非系统性风险】;
$E\left(R_i\right)$ is the expected return on stock $\mathrm{i}$;
The nonsystematic components of returns $e_i$, are assumed to be uncorrelated among themselves and uncorrelated with the factor F.

The expected value of $e_i$ for any well-diversified portfolio is zero, and its variance also is effectively zero. We conclude that for a well-diversified portfolio, for all practical purposes
$$
R_i=E\left(R_i\right)+\beta_i F
$$

Multifactor Models

只讨论把系统性风险分解，firm-specific risk 不能分解。

Models that allow for several factors can provide better descriptions of security returns.

Multifactor models of security returns can be used to measure and manage exposure to each of many economy-wide factors such as business-cycle risk, interest or inflation rate risk, energy price risk,and so on.

Factor models are tools that allow us to describe and quantify the different factors that affect the rate of return on a security during any time period.

The equation for multifactor model for stock $i$

$R_i=E\left(R_i\right)+\beta_{i, G D P} G D P+\beta_{i, I R} I R+e_i$

$R_i$ =return on stock $\mathrm{i}$
$E(R_i)$ =expected rate of return for stock $i$
$\beta_{i, G D P}$= GDP factor beta for stock $i$
$\beta_{i, I R}$= interest rate factor beta for stock $i$
GDP = deviation of GDP factor from its expected value
IR = deviation of interest rate factor from its expected value
$e_i$ =firm-specific return for stock $i$

Arbitrage Pricing Theory

原理是把系统性风险进一步拆分。

Arbitrage【套利】 pricing theory (APT) is a general theory of asset pricing that holds that the expected return of a financial asset can be modeled as a linear function of various macro-economic factors or theoretical market indices , where sensitivity to changes in each factor is represented by a factor specific beta coefficient.

Assumptions

Asset returns can be explained by systemic factors.
By using diversification, investors can eliminate specific risk from their portfolios.

和多因素模型相比较而言，没有非系统性风险。

There are no arbitrage opportunities among well-diversified portfolios. If any arbitrage opportunities were to exist, investors would exploit them away

Arbitrage, Risk Arbitrage, and Equilibrium

CAPM’s argument: If a security is mispriced, then investors will tilt their portfolios toward the underpriced and away from the overpriced securities,each by a relatively small dollar amount.

投资思路：计算required return；比较；决定是否投资

tip：【价值投资学派】【要先计算必要回报率】

APT’s implication: a few investors who identify an arbitrage opportunity will mobilize large dollar amounts and quickly restore equilibrium.

投资思路：寻找两个相同的资产【系统性风险相同】；判断有无套利机会；决定是否投资

tip：【套利学派】【如果两个相同的资产价格不同，那就通过倒买倒卖获利】

Arbitrageur often refers to a professional searching for mispriced securities in specific areas such as merger-target stocks

Risk Premiums

再muti-factor中，称为deviation

The risk premiums are derived as follows:
Create factor portfolios. Each factor portfolio is a well-diversified portfolio that has a beta equal to one for a single risk factor, and betas equal to zero on the remaining factors.
Derive returns for each factor portfolio. For instance, define 𝐸(𝑅𝑖) as the expected return on Factor Portfolio i.
Calculate risk premiums for each factor portfolio

Empirical Work Preference【经验之谈】

APT approach is more popular than CAPM in empirical works for the following reasons:

CAPM is a special case of APT. CAPM is a one-factor model while APT is a multi-factor model.
APT better helps facilitate risk analyses than CAPM. Consider the case of a portfolio with $\mathrm{n}=50$ different equities:
- CAPM requires a full volatility and correlation matrix, which means there will be a total of $50+\frac{50^2-50}{2}=1275$ calculations.
- With a multi-factor model, for example, if the 50 equities are categorized into 3 factors, the number of calculations will be $3 \times 50+\frac{3^2-3}{2}=153$

Fama-French Three-Factor Model

The Fama-French three-factor model incorporates the following systematic factors:
$$
R_{i t}-r_f=\alpha_i+\beta_{i M}\left(R_{M t}-r_f\right)+\beta_{i, S M B} S M B_t+\beta_{i, H M L} H M L_t+e_{i t}
$$

$R_{M t}-r_f$市场因子market factor

SMB：small minus big factor;小盘股的收益率高于大盘股【小盘股定价投资者较少，错误定价的可能性高】。

HML：high minus Low factor；长期来看，价值股收益高于成长股，应该卖出成长股买入价值股；

$\mathrm{SMB}=$ Small minus big (the return of a portfolio of small stocks - return on a portfolio of large stocks)
$\mathrm{HML}=$ High minus low (the return of a portfolio of stocks with a high book-to-market ratio【账面价值/市面价值】 - return on a portfolio of stocks with a low book to-market ratio)

F&F have observed: firms with high ratios of book-to-market value are more likely to be in financial distress and that small stocks may be more sensitive to changes in business conditions. Thus, these variables may capture sensitivity to risk factors in the macro economy.

Factor Analysis in Hedging Exposure

Idiosyncratic (specific) risk can theoretically be eliminated through diversification, but this is not true for systematic risk.

Challenges to the hedge:

The selection of appropriate systematic factors
The frequency of adjusting a hedge【对冲是有风险的】
Model risk

B:净投资额为0【空手套白狼】