Assets Pricing 资产定价（四）

Lec 5 APT Model-Multi Factor Model 套利定价理论-多因子模型

Problem of CAPM

• Since CAPM is based on historical data it is most appropriate when conditions are relatively stable. If conditions change, the beta coefficient estimated from the sample may not correspond to the likely future value. CAPM是基于历史数据的，因此在条件稳定的时候最准确。如果条件改变，从历史数据中估计出的beta系数可能于未来值不符，从而产生定价偏差。
• The efficacy of CAPM tests is conditional on the efficiency of the market portfolio. The index turns out to be ex-post efficient, if every asset is falling on the security market line. CAPM测试的有效性取决于市场组合的效率。如果所有资产都落在证券市场线上，那么该指数就具有事后效率。

Basic idea of ATP

The idea of arbitrage underlies the APT. Arbitrage ensures that the same “bundle” of systematic risks sells for the same price.

Expected and Unexpected return

${\overline{R}}_i$: is expected return, is not a random variable and is not the source of risk.

$u_i$: is unexpected shocks and the risk raises from it. The characteristic of $u_i$ will influences ${\overline{R}}_i$

$R_i={\overline{R}}_i+u_i$

Systematic and unsystematic risk

$u_i=m_i+{ϵ}_i$

• Systematic risk for asset $i$, $m_i$, is then related to unexpected economy-wide shocks (such as unanticipated fluctuations in GNP, inflation, the terms of trade or real interest rates):
  $$m_i={\beta }_{iY}{F}_Y+{\beta }_{i\pi }{F}_{\pi }+{\beta }_{iX}{F}_X+{\beta }_{ir}{F}_r$$
  Systematic risk $m_i$ cannot be eliminated by holding a diversified portfolio.

• In a $k$-factor model we simply use a numerical index to denote each of the systematic risks affecting returns on stock $i$:
  $$R_i={\overline{R}}_i+u_i={\overline{R}}_i+{\beta }_{i1}{F}_1+{\beta }_{i2}{F}_2+\cdots +{\beta }_{ik}{F}_k+{ϵ}_i$$
  Unsystematic risk ${ϵ}_i$ can be eliminated by holding a diversified portfolio. For market portfolio ${x\text{x}x}_M$ with $N$ stocks, ${ϵ}_i$ satisfies
  $$Var(\sum _{i=1}^N{x}_i{ϵ}_i)=0$$

Arbitrage also implies the expected return ${\overline{R}}_i$, on a stock $i$ will depend on the ${\beta }_{ij}$ coefficients showing how systematic unexpected shocks $F_j$ affect unexpected returns $u_i=R_i-{\overline{R}}_i$.

And the equilibrium returns ${\overline{R}}_{F_j}$, for bearing “one unit” of each type of systematic shock $F_j$. So we have:
$${\overline{R}}_i=r+{\beta }_{i1}[{\overline{R}}_{F_1}-r]+{\beta }_{i2}[{\overline{R}}_{F_2}-r]+\cdots +{\beta }_{ik}[{\overline{R}}_{F_k}-r]$$
Plug in, we get the Actual Returns under APT:
$$\begin{array}{rl}{R}_i& ={\overline{R}}_i+{\beta }_{i1}{F}_1+{\beta }_{i2}{F}_2+\cdots +{\beta }_{ik}{F}_k+{ϵ}_i\\ & =r+\sum _{j=1}^k{\beta }_{ij}[{\overline{R}}_{F_j}-r]+\sum _{j=1}^k{\beta }_{ij}{F}_j\end{array}$$

• $\beta$ for a stock has two effects on its actual returns

Question:

(1) How do CAPM and APT models explain systematic risk and unsystematic risk for an asset (or stock) in the stock market respectively?

(2) Are CAPM and APT consistent (or compatible) ? Please show it.

(3) Compare the advantages of APT model over CAPM

(1)

CAPM
$$R_i-r={\alpha }_i+{\beta }_i(R_M-r)+{ϵ}_i$$

• Term ${\beta }_i(R_M-r)$ corresponds to the systematic risk, which cannot be diversified.

• The error term ${ϵ}_i$ corresponds to the unsystematic risk, which satisfies $Cov({ϵ}_i,R_M)=0$.

Hence the following decomposition of total risk as measured by the variance is possible:
$$Var(R_i)={\beta }_i^{2}Var(R_M)+Var({ϵ}_i)$$
APT
$$R_i={\overline{R}}_i+u_i={\overline{R}}_i+m_i+{ϵ}_i$$

• Systematic risk for asset $i$, $m_i$, is then related to unexpected economy-wide shocks.

  Systematic risk $m_i$ cannot be eliminated by holding a diversified portfolio.

  In a $k$-factor model we simply use a numerical index to denote each of the systematic risks affecting returns on stock $i$:

$$R_i={\overline{R}}_i+u_i={\overline{R}}_i+{\beta }_{i1}{F}_1+{\beta }_{i2}{F}_2+\cdots +{\beta }_{ik}{F}_k+{ϵ}_i$$

• Unsystematic risk ${ϵ}_i$ can be eliminated by holding a diversified portfolio. For market portfolio ${x\text{x}x}_M$ with $N$ stocks, ${ϵ}_i$ satisfies
  $$Var(\sum _{i=1}^N{x}_i{ϵ}_i)=0$$

(2)

Consistent.

The Actual return for asset (or stock) $i$ is consistent:

APT holds the equation for expected return:
$${\overline{R}}_i=r+\sum _{i=1}^k{\beta }_{ij}[{\overline{R}}_{F_j}-r]$$
We think CAPM as a special case where there is just one “factor”, namely $F_M$ (the unanticipated return on a broad-based market portfolio such as S&P 500):
$${\overline{R}}_i=r+{\beta }_{iM}[{\overline{R}}_M-r]$$
Actual returns in APT model holds the equation:
$$\begin{gathered} R_i={\overline{R}}_i+\sum _{j=1}^k{\beta }_{ij}{F}_j+{ϵ}_i \\ =r+\sum _{i=1}^k{\beta }_{ij}[{\overline{R}}_{F_j}-r]+\sum _{j=1}^k{\beta }_{ij}{F}_j+{ϵ}_i \end{gathered}$$
This equation extends the CAPM in this case to apply to actual return as opposed to expected return:
$$\begin{array}{rl}{R}_i& ={\overline{R}}_i+{\beta }_{iM}{F}_M+{ϵ}_i\\ & =r+{\beta }_{iM}[{\overline{R}}_M-r]+{\beta }_{iM}{F}_M+{ϵ}_i\\ & =r+{\beta }_{iM}[{\overline{R}}_M-r]+{\beta }_{iM}[R_M-{\overline{R}}_M]+{ϵ}_i\\ & =r+{\beta }_{iM}[R_M-r]+{ϵ}_i\end{array}$$
With ${\beta }_{iM}={\beta }_i$ and $r=R_f$ in CAPM, this equation $R_i=R_f+{\beta }_i+[R_M-R_f]+{ϵ}_i$ is just the regression equation used to implement in CAPM.

The Expected return for asset (or stock) $i$ is consistent:

Conversely, if CAPM is valid, the expected return of asset $i$ is given by:
$$\begin{gathered} E(R_i)=R_f+{\beta }_i[E(R_M)-R_f] \\ i.e.{\overline{R}}_i=r+{\beta }_{iM}[{\overline{R}}_M-r] \end{gathered}$$
The expected return for asset $i$ according to APT is
$${\overline{R}}_i=r+\sum _{i=1}^k{\beta }_{ij}[{\overline{R}}_{F_j}-r]$$
If CAPM holds, the expected return ${\overline{R}}_{F_j}$ will satisfy:
$${\overline{R}}_{F_j}=r+{\beta }_{F_{j}M}[{\overline{R}}_M-r]$$
Plug in:
$$\begin{array}{rl}{\overline{R}}_i& =r+\sum _{i=1}^k{\beta }_{ij}{\beta }_{F_{j}M}[{\overline{R}}_M-r]\\ & =r+({\overline{R}}_M-r)\sum _{i=1}^k{\beta }_{ij}{\beta }_{F_{j}M}\\ & =r+({\overline{R}}_M-r){\beta }_{iM}\end{array}$$
Thus, as long as $\sum _{i=1}^k{\beta }_{ij}{\beta }_{F_{j}M}={\beta }_{iM}$, ${\overline{R}}_i=r+{\beta }_{iM}({\overline{R}}_M-r)$. It’s consistent to the expected return on asset $i$ given by CAPM.

In conclusion, a $k$ factor APT model can be consistent with CAPM if the factors are priced right. CAPM is a special case of APT model.

(3)

⭐Advantages

• In contrast to CAPM, the APT:

  • allows for multiple risk factors that can vary in relative importance over time; 随时间变化的多因子
  • can be used to test information about the types of shocks that lead to market risk; 对引起市场风险的不同类型冲击进行信息检验
  • may better predict risk premiums for a project that combines the risks embodied in existing projects. 对现存研究中的风险因素进行组合研究来更好地预测风险溢价

• On the other hand:

  • CAPM is more closely related to the underlying economic choice - namely, the trade-off between risk and return.平衡风险和收益
  • The development of the CAPM enables us to derive the risk return separation theorem.推出风险回报分离定理
  • The discussion of efficient sets elucidates (shows) the basic trade-off.有效集阐明了基础平衡

APT vs CAPM (differences)

• APT makes no assumption about empirical distribution of asset returns 对资产回报的分布没有经验假设

• No assumption of individual’s utility function 没有假设个体效用函数

• More than 1 factor 多因子

• It is for any subset of securities 证券的子集

• No special role for the market portfolio in APT. 没有市场投资组合的角色

• Can be easily extended to a multi-period framework. 容易拓展到多阶段的理论框架

Driving Arbitrage Pricing Theory Model

Assuming that the rate of return on each of the $n$ security is a linear function of $k$ factors:
$$R_i=E(R_i)+b_{i1}{F}_1+b_{i2}{F}_2+\cdots +b_{ik}{F}_k+{ϵ}_i$$
where

• $i=1,2,\dots ,n$

• $R_i$ and $E(R_i)$ are the random and expected return rates on the $i^{th}$ asset;

• $b_{ik}$ is the sensitivity of the $i^{th}$ asset’s return to the $k^{th}$ factor;

• $F_k$ is the mean zero $k^{th}$ factor common to the returns of all assets, $Cov(F_k,F_h)=0(k\ne h)$, $k$ factors are not co-related;

• ${ϵ}_i$ is a random zero mean noise term for the $i^{th}$ asset.

Construct an arbitrage portfolio $P$ using the above $n$ assets:

• No wealth:
  $$\sum _{i=1}^n{w}_i=0\text{or}{e}^{T}w=w^{T}e=0$$
  $w_i$ is the weight of security $i$ in the portfolio, ${w\text{w}w}^T=(w_1,w_2,\dots ,w_n),{e\text{e}e}^T=(1,1,\dots ,1)$.

• Having no risk and earning no return on average.

If $P$ is well-diversified portfolio, the return of the arbitrage portfolio $R_P$ is independent of individual risk of each security:
$$R_P=\sum _{i=1}^n{w}_i{R}_i=\sum _{i=1}^n{w}_{i}E(R_i)+\sum _{i=1}^n{w}_i{b}_{i1}{F}_1+\sum _{i=1}^n{w}_i{b}_{i2}{F}_2+\cdots +\sum _{i=1}^n{w}_i{b}_{ik}{F}_k+\sum _{i=1}^n{w}_i{ϵ}_i$$
To obtain a riskless arbitrage portfolio, one needs to eliminate both diversifiable and nondiversifiable risks:
$$\begin{gathered} w_i\approx \frac{1}{n},n\to \infty \Rightarrow \sum _{i=1}^n{w}_i{b}_{ik}=0\text{for all factors.} \\ \Rightarrow R_P=\sum _{i=1}^n{w}_{i}E(R_i)=0 \end{gathered}$$

• The deterministic return of the portfolio must equals to 0, otherwise there exists arbitrage opportunity.

We have vectors and matrix:
$$\begin{gathered} {w\text{w}w}^T=(w_1,w_2,\dots ,w_n) \\ {\mu \text{μ}\mu }^T=(E(R_1),E(R_2),\dots ,E(R_n)) \\ b\text{b}b=\left[\begin{array}{ccccc}1& b_{11}& b_{12}& \cdots & b_{1k}\\ 1& b_{21}& b_{22}& \cdots & b_{2k}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& b_{n1}& b_{n2}& \cdots & b_{nk}\end{array}\right] \end{gathered}$$
There exists a set of $k+1$ coefficients ${\lambda }_0,{\lambda }_1,\dots ,{\lambda }_k$, i.e. ${\lambda \text{λ}\lambda }^T=({\lambda }_0,{\lambda }_1,\dots ,{\lambda }_k)$ such that

$$\begin{array}{r}b\text{b}b\lambda \text{λ}\lambda =\left[\begin{array}{ccccc}1& b_{11}& b_{12}& \cdots & b_{1k}\\ 1& b_{21}& b_{22}& \cdots & b_{2k}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& b_{n1}& b_{n2}& \cdots & b_{nk}\end{array}\right]\left[\begin{array}{c}{\lambda }_0\\ {\lambda }_1\\ ⋮\\ {\lambda }_k\end{array}\right]=\left[\begin{array}{c}{\lambda }_0+{\lambda }_1{b}_{11}+\cdots +{\lambda }_k{b}_{1k}\\ {\lambda }_0+{\lambda }_1{b}_{21}+\cdots +{\lambda }_k{b}_{2k}\\ ⋮\\ {\lambda }_0+{\lambda }_1{b}_{n1}+\cdots +{\lambda }_k{b}_{nk}\end{array}\right]=\left[\begin{array}{c}E(R_1)\\ E(R_2)\\ ⋮\\ E(R_n)\end{array}\right]=\mu \text{μ}\mu \end{array}$$

$$\begin{array}{rl}{w\text{w}w}^T\mu \text{μ}\mu & =(w_1,w_2,\dots ,w_n)\left[\begin{array}{c}{\lambda }_0+{\lambda }_1{b}_{11}+\cdots +{\lambda }_k{b}_{1k}\\ {\lambda }_0+{\lambda }_1{b}_{21}+\cdots +{\lambda }_k{b}_{2k}\\ ⋮\\ {\lambda }_0+{\lambda }_1{b}_{n1}+\cdots +{\lambda }_k{b}_{nk}\end{array}\right]\\ & =w_1({\lambda }_0+{\lambda }_1{b}_{11}+\cdots +{\lambda }_k{b}_{1k})+w_2({\lambda }_0+{\lambda }_1{b}_{21}+\cdots +{\lambda }_k{b}_{2k})+\cdots +w_n({\lambda }_0+{\lambda }_1{b}_{n1}+\cdots +{\lambda }_k{b}_{nk})\\ & ={\lambda }_0\sum _{i=1}^n{w}_i+{\lambda }_1\sum _{i=1}^n{w}_i{b}_{i1}+\cdots +{\lambda }_k\sum _{i=1}^n{w}_i{b}_{ik}=0\end{array}$$

We have $E(R_i)={\lambda }_0+{\lambda }_1{b}_{i1}+\cdots +{\lambda }_k{b}_{ik}$ satisfying $\sum _{i=1}^n{w}_{i}E(R_i)={w\text{w}w}^T\mu \text{μ}\mu =0$. See ${\lambda }_0$ as risk-free return, i.e. ${\lambda }_0=R_f$, so we have
$$E(R_i)=R_f+{\lambda }_1{b}_{i1}+\cdots +{\lambda }_k{b}_{ik}$$
Specifically, if one factor portfolio only depends on factor 1 ($F_1$), then
$$E(R_1)=R_f+1\cdot {\lambda }_1\Rightarrow {\lambda }_1=E(R_1)-R_f$$
Similarly,
$$\begin{gathered} {\lambda }_2=E(R_2)-R_f \\ \dots \\ {\lambda }_k=E(R_k)-R_f \end{gathered}$$
Therefore,
$$E(R_i)=R_f+b_{i1}[E(R_1)-R_f]+\cdots +b_{ik}[E(R_k)-R_f]$$