Assets Pricing 资产定价（四）

文章目录

Lec 5 APT Model-Multi Factor Model 套利定价理论-多因子模型
- Problem of CAPM
- Basic idea of ATP
- - Expected and Unexpected return
  - Systematic and unsystematic risk
  - Driving Arbitrage Pricing Theory Model

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

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

uiu_iui: is unexpected shocks and the risk raises from it. The characteristic of uiu_iui will influences R‾i\overline{R}_iRi

Ri=R‾i+uiR_i=\overline{R}_i+u_iRi=Ri+ui

Systematic and unsystematic risk

ui=mi+ϵiu_i=m_i+\epsilon_iui=mi+ϵi

Systematic risk for asset iii, mim_imi, is then related to unexpected economy-wide shocks (such as unanticipated fluctuations in GNP, inflation, the terms of trade or real interest rates):
mi=βiYFY+βiπFπ+βiXFX+βirFrm_i=\beta_{iY}F_Y+\beta_{i\pi}F_{\pi}+\beta_{iX}F_X+\beta_{ir}F_r mi=βiYFY+βiπFπ+βiXFX+βirFr
Systematic risk mim_imi cannot be eliminated by holding a diversified portfolio.
In a kkk-factor model we simply use a numerical index to denote each of the systematic risks affecting returns on stock iii:
Ri=R‾i+ui=R‾i+βi1F1+βi2F2+⋯+βikFk+ϵiR_i=\overline {R}_i+u_i=\overline {R}_i+\beta_{i1}F_1+\beta_{i2}F_2+\dots+\beta_{ik}F_k+\epsilon_i Ri=Ri+ui=Ri+βi1F1+βi2F2+⋯+βikFk+ϵi
Unsystematic risk ϵi\epsilon_iϵi can be eliminated by holding a diversified portfolio. For market portfolio xM\pmb x_MxxxM with NNN stocks, ϵi\epsilon_iϵi satisfies
Var(∑i=1Nxiϵi)=0Var(\displaystyle\sum_{i=1}^Nx_i\epsilon_i)=0 Var(i=1∑Nxiϵi)=0

Arbitrage also implies the expected return R‾i\overline R_iRi, on a stock iii will depend on the βij\beta_{ij}βij coefficients showing how systematic unexpected shocks FjF_jFj affect unexpected returns ui=Ri−R‾iu_i=R_i-\overline{R}_iui=Ri−Ri.

And the equilibrium returns R‾Fj\overline R_{F_j}RFj, for bearing “one unit” of each type of systematic shock FjF_jFj. So we have:
R‾i=r+βi1[R‾F1−r]+βi2[R‾F2−r]+⋯+βik[R‾Fk−r]\overline {R}_i=r+\beta_{i1}[\overline R_{F_1}-r]+\beta_{i2}[\overline R_{F_2}-r]+\dots+\beta_{ik}[\overline R_{F_k}-r] Ri=r+βi1[RF1−r]+βi2[RF2−r]+⋯+βik[RFk−r]
Plug in, we get the Actual Returns under APT:
Ri=R‾i+βi1F1+βi2F2+⋯+βikFk+ϵi=r+∑j=1kβij[R‾Fj−r]+∑j=1kβijFj\begin{aligned} R_i&=\overline {R}_i+\beta_{i1}F_1+\beta_{i2}F_2+\dots+\beta_{ik}F_k+\epsilon_i \\ &=r+\displaystyle\sum_{j=1}^k\beta_{ij}[\overline R_{F_j}-r]+\displaystyle\sum_{j=1}^k\beta_{ij}F_j \end{aligned} Ri=Ri+βi1F1+βi2F2+⋯+βikFk+ϵi=r+j=1∑kβij[RFj−r]+j=1∑kβijFj

β\betaβ 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
Ri−r=αi+βi(RM−r)+ϵiR_i-r=\alpha_i+\beta_i(R_M-r)+\epsilon_i Ri−r=αi+βi(RM−r)+ϵi

Term βi(RM−r)\beta_i(R_M-r)βi(RM−r) corresponds to the systematic risk, which cannot be diversified.

The error term ϵi\epsilon_iϵi corresponds to the unsystematic risk, which satisfies Cov(ϵi,RM)=0Cov(\epsilon_i,R_M)=0Cov(ϵi,RM)=0.

Hence the following decomposition of total risk as measured by the variance is possible:
Var(Ri)=βi2Var(RM)+Var(ϵi)Var(R_i)=\beta_i^2Var(R_M)+Var(\epsilon_i) Var(Ri)=βi2Var(RM)+Var(ϵi)
APT
Ri=R‾i+ui=R‾i+mi+ϵiR_i=\overline {R}_i+u_i=\overline {R}_i+m_i+\epsilon_i Ri=Ri+ui=Ri+mi+ϵi

Systematic risk for asset iii, mim_imi, is then related to unexpected economy-wide shocks.

Systematic risk mim_imi cannot be eliminated by holding a diversified portfolio.

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

Ri=R‾i+ui=R‾i+βi1F1+βi2F2+⋯+βikFk+ϵiR_i=\overline {R}_i+u_i=\overline {R}_i+\beta_{i1}F_1+\beta_{i2}F_2+\dots+\beta_{ik}F_k+\epsilon_i Ri=Ri+ui=Ri+βi1F1+βi2F2+⋯+βikFk+ϵi

Unsystematic risk ϵi\epsilon_iϵi can be eliminated by holding a diversified portfolio. For market portfolio xM\pmb x_MxxxM with NNN stocks, ϵi\epsilon_iϵi satisfies
Var(∑i=1Nxiϵi)=0Var(\displaystyle\sum_{i=1}^Nx_i\epsilon_i)=0 Var(i=1∑Nxiϵi)=0

(2)

Consistent.

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

APT holds the equation for expected return:
R‾i=r+∑i=1kβij[R‾Fj−r]\overline{R}_i=r+\displaystyle\sum_{i=1}^k\beta_{ij}[\overline R_{F_j}-r] Ri=r+i=1∑kβij[RFj−r]
We think CAPM as a special case where there is just one “factor”, namely FMF_MFM (the unanticipated return on a broad-based market portfolio such as S&P 500):
R‾i=r+βiM[R‾M−r]\overline{R}_i=r+\beta_{iM}[\overline R_M-r] Ri=r+βiM[RM−r]
Actual returns in APT model holds the equation:
Ri=R‾i+∑j=1kβijFj+ϵi=r+∑i=1kβij[R‾Fj−r]+∑j=1kβijFj+ϵiR_i=\overline{R}_i+\displaystyle\sum_{j=1}^k\beta_{ij}F_j+\epsilon_i\\ =r+\displaystyle\sum_{i=1}^k\beta_{ij}[\overline R_{F_j}-r]+\displaystyle\sum_{j=1}^k\beta_{ij}F_j+\epsilon_i Ri=Ri+j=1∑kβijFj+ϵi=r+i=1∑kβij[RFj−r]+j=1∑kβijFj+ϵi
This equation extends the CAPM in this case to apply to actual return as opposed to expected return:
Ri=R‾i+βiMFM+ϵi=r+βiM[R‾M−r]+βiMFM+ϵi=r+βiM[R‾M−r]+βiM[RM−R‾M]+ϵi=r+βiM[RM−r]+ϵi\begin{aligned} R_i&=\overline{R}_i+\beta_{iM}F_M+\epsilon_i\\ &=r+\beta_{iM}[\overline R_M-r]+\beta_{iM}F_M+\epsilon_i\\ &=r+\beta_{iM}[\overline R_M-r]+\beta_{iM}[R_M-\overline R_M]+\epsilon_i\\ &=r+\beta_{iM}[R_M-r]+\epsilon_i \end{aligned} Ri=Ri+βiMFM+ϵi=r+βiM[RM−r]+βiMFM+ϵi=r+βiM[RM−r]+βiM[RM−RM]+ϵi=r+βiM[RM−r]+ϵi
With βiM=βi\beta_{iM}=\beta_iβiM=βi and r=Rfr= R_fr=Rf in CAPM, this equation Ri=Rf+βi+[RM−Rf]+ϵiR_i=R_f+\beta_i+[R_M-R_f]+\epsilon_iRi=Rf+βi+[RM−Rf]+ϵi is just the regression equation used to implement in CAPM.

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

Conversely, if CAPM is valid, the expected return of asset iii is given by:
E(Ri)=Rf+βi[E(RM)−Rf]i.e.R‾i=r+βiM[R‾M−r]E(R_i)=R_f+\beta_i[E(R_M)-R_f]\\ i.e.\ \overline R_i=r+\beta_{iM}[\overline R_M-r] E(Ri)=Rf+βi[E(RM)−Rf]i.e. Ri=r+βiM[RM−r]
The expected return for asset iii according to APT is
R‾i=r+∑i=1kβij[R‾Fj−r]\overline{R}_i=r+\displaystyle\sum_{i=1}^k\beta_{ij}[\overline R_{F_j}-r] Ri=r+i=1∑kβij[RFj−r]
If CAPM holds, the expected return R‾Fj\overline R_{F_j}RFj will satisfy:
R‾Fj=r+βFjM[R‾M−r]\overline R_{F_j}=r+\beta_{F_jM}[\overline R_M-r] RFj=r+βFjM[RM−r]
Plug in:
R‾i=r+∑i=1kβijβFjM[R‾M−r]=r+(R‾M−r)∑i=1kβijβFjM=r+(R‾M−r)βiM\begin{aligned} \overline{R}_i&=r+\displaystyle\sum_{i=1}^k\beta_{ij}\beta_{F_jM}[\overline R_M-r]\\ &=r+(\overline R_M-r)\displaystyle\sum_{i=1}^k\beta_{ij}\beta_{F_jM}\\ &=r+(\overline R_M-r)\beta_{iM} \end{aligned} Ri=r+i=1∑kβijβFjM[RM−r]=r+(RM−r)i=1∑kβijβFjM=r+(RM−r)βiM
Thus, as long as ∑i=1kβijβFjM=βiM\displaystyle\sum_{i=1}^k\beta_{ij}\beta_{F_jM}=\beta_{iM}i=1∑kβijβFjM=βiM, R‾i=r+βiM(R‾M−r)\overline{R}_i=r+\beta_{iM}(\overline R_M-r)Ri=r+βiM(RM−r). It’s consistent to the expected return on asset iii given by CAPM.

In conclusion, a kkk 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 nnn security is a linear function of kkk factors:
Ri=E(Ri)+bi1F1+bi2F2+⋯+bikFk+ϵiR_i=E(R_i)+b_{i1}F_1+b_{i2}F_2+\dots+b_{ik}F_k+\epsilon_i Ri=E(Ri)+bi1F1+bi2F2+⋯+bikFk+ϵi
where

i=1,2,…,ni=1,2,\dots ,ni=1,2,…,n
RiR_iRi and E(Ri)E(R_i)E(Ri) are the random and expected return rates on the ithi^{th}ith asset;
bikb_{ik}bik is the sensitivity of the ithi^{th}ith asset’s return to the kthk^{th}kth factor;
FkF_kFk is the mean zero kthk^{th}kth factor common to the returns of all assets, Cov(Fk,Fh)=0(k≠h)Cov(F_k,F_h)=0\ (k\neq h)Cov(Fk,Fh)=0 (k=h), kkk factors are not co-related;
ϵi\epsilon_iϵi is a random zero mean noise term for the ithi^{th}ith asset.

Construct an arbitrage portfolio PPP using the above nnn assets:

No wealth:
∑i=1nwi=0or eTw=wTe=0\displaystyle \sum_{i=1}^{n}w_i=0\ \ \ \text{or}\ \ \ e^Tw=w^Te=0 i=1∑nwi=0 or eTw=wTe=0
wiw_iwi is the weight of security iii in the portfolio, wT=(w1,w2,…,wn),eT=(1,1,…,1)\pmb w^T=(w_1,w_2,\dots,w_n), \pmb e^T=(1,1,\dots,1)wwwT=(w1,w2,…,wn),eeeT=(1,1,…,1).
Having no risk and earning no return on average.

If PPP is well-diversified portfolio, the return of the arbitrage portfolio RPR_PRP is independent of individual risk of each security:
RP=∑i=1nwiRi=∑i=1nwiE(Ri)+∑i=1nwibi1F1+∑i=1nwibi2F2+⋯+∑i=1nwibikFk+∑i=1nwiϵiR_P=\displaystyle \sum_{i=1}^{n}w_i R_i =\displaystyle \sum_{i=1}^{n}w_iE(R_i)+\displaystyle \sum_{i=1}^{n}w_ib_{i1}F_1+\displaystyle \sum_{i=1}^{n}w_ib_{i2}F_2+\dots+\displaystyle \sum_{i=1}^{n}w_ib_{ik}F_k+\displaystyle \sum_{i=1}^{n}w_i\epsilon_i RP=i=1∑nwiRi=i=1∑nwiE(Ri)+i=1∑nwibi1F1+i=1∑nwibi2F2+⋯+i=1∑nwibikFk+i=1∑nwiϵi
To obtain a riskless arbitrage portfolio, one needs to eliminate both diversifiable and nondiversifiable risks:
wi≈1n,n→∞⇒∑i=1nwibik=0for all factors.⇒RP=∑i=1nwiE(Ri)=0w_i\approx\frac{1}{n},n\to\infty\Rightarrow\displaystyle \sum_{i=1}^{n}w_ib_{ik}=0\ \ \ \text{for all factors.}\\ \Rightarrow R_P=\displaystyle \sum_{i=1}^{n}w_iE(R_i)=0 wi≈n1,n→∞⇒i=1∑nwibik=0 for all factors.⇒RP=i=1∑nwiE(Ri)=0

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

We have vectors and matrix:
wT=(w1,w2,…,wn)μT=(E(R1),E(R2),…,E(Rn))b=[1b11b12⋯b1k1b21b22⋯b2k⋮⋮⋮⋱⋮1bn1bn2⋯bnk]\pmb w^T=(w_1,w_2,\dots,w_n)\\ \pmb \mu^T=(E(R_1),E(R_2),\dots,E(R_n))\\ \pmb b = \left[ \begin{matrix} 1& b_{11} &b_{12} &\cdots &b_{1k}\\1&b_{21} &b_{22} &\cdots &b_{2k}\\\vdots &\vdots &\vdots &\ddots &\vdots\\1&b_{n1} &b_{n2} &\cdots &b_{nk}\end{matrix} \right] wwwT=(w1,w2,…,wn)μμμT=(E(R1),E(R2),…,E(Rn))bbb=⎣⎢⎢⎢⎡11⋮1b11b21⋮bn1b12b22⋮bn2⋯⋯⋱⋯b1kb2k⋮bnk⎦⎥⎥⎥⎤
There exists a set of k+1k+1k+1 coefficients λ0,λ1,…,λk\lambda_0,\lambda_1,\dots,\lambda_kλ0,λ1,…,λk, i.e. λT=(λ0,λ1,…,λk)\pmb \lambda^T=(\lambda_0,\lambda_1,\dots,\lambda_k)λλλT=(λ0,λ1,…,λk) such that

bλ=[1b11b12⋯b1k1b21b22⋯b2k⋮⋮⋮⋱⋮1bn1bn2⋯bnk][λ0λ1⋮λk]=[λ0+λ1b11+⋯+λkb1kλ0+λ1b21+⋯+λkb2k⋮λ0+λ1bn1+⋯+λkbnk]=[E(R1)E(R2)⋮E(Rn)]=μ\begin{aligned} \pmb b\pmb \lambda=\left[ \begin{matrix} 1& b_{11} &b_{12} &\cdots &b_{1k}\\1&b_{21} &b_{22} &\cdots &b_{2k}\\\vdots &\vdots &\vdots &\ddots &\vdots\\1&b_{n1} &b_{n2} &\cdots &b_{nk}\end{matrix} \right] \left[ \begin{matrix} \lambda_0\\\lambda_1\\\vdots \\\lambda_k\end{matrix} \right]= \left[ \begin{matrix} \lambda_0+\lambda_1b_{11}+\dots+\lambda_kb_{1k}\\\lambda_0+\lambda_1b_{21}+\dots+\lambda_kb_{2k}\\\vdots \\\lambda_0+\lambda_1b_{n1}+\dots+\lambda_kb_{nk}\end{matrix} \right]= \left[ \begin{matrix} E(R_1)\\E(R_2)\\\vdots \\E(R_n)\end{matrix} \right]= \pmb \mu \end{aligned} bbbλλλ=⎣⎢⎢⎢⎡11⋮1b11b21⋮bn1b12b22⋮bn2⋯⋯⋱⋯b1kb2k⋮bnk⎦⎥⎥⎥⎤⎣⎢⎢⎢⎡λ0λ1⋮λk⎦⎥⎥⎥⎤=⎣⎢⎢⎢⎡λ0+λ1b11+⋯+λkb1kλ0+λ1b21+⋯+λkb2k⋮λ0+λ1bn1+⋯+λkbnk⎦⎥⎥⎥⎤=⎣⎢⎢⎢⎡E(R1)E(R2)⋮E(Rn)⎦⎥⎥⎥⎤=μμμ

wTμ=(w1,w2,…,wn)[λ0+λ1b11+⋯+λkb1kλ0+λ1b21+⋯+λkb2k⋮λ0+λ1bn1+⋯+λkbnk]=w1(λ0+λ1b11+⋯+λkb1k)+w2(λ0+λ1b21+⋯+λkb2k)+⋯+wn(λ0+λ1bn1+⋯+λkbnk)=λ0∑i=1nwi+λ1∑i=1nwibi1+⋯+λk∑i=1nwibik=0\begin{aligned} \pmb w^T\pmb \mu&=(w_1,w_2,\dots,w_n)\left[ \begin{matrix} \lambda_0+\lambda_1b_{11}+\dots+\lambda_kb_{1k}\\\lambda_0+\lambda_1b_{21}+\dots+\lambda_kb_{2k}\\\vdots \\\lambda_0+\lambda_1b_{n1}+\dots+\lambda_kb_{nk}\end{matrix} \right]\\ &=w_1(\lambda_0+\lambda_1b_{11}+\dots+\lambda_kb_{1k})+w_2(\lambda_0+\lambda_1b_{21}+\dots+\lambda_kb_{2k})+\dots+w_n(\lambda_0+\lambda_1b_{n1}+\dots+\lambda_kb_{nk})\\ &=\lambda_0\displaystyle\sum_{i=1}^n w_i+\lambda_1\displaystyle\sum_{i=1}^n w_ib_{i1}+\dots+\lambda_k\displaystyle\sum_{i=1}^n w_ib_{ik}=0 \end{aligned} wwwTμμμ=(w1,w2,…,wn)⎣⎢⎢⎢⎡λ0+λ1b11+⋯+λkb1kλ0+λ1b21+⋯+λkb2k⋮λ0+λ1bn1+⋯+λkbnk⎦⎥⎥⎥⎤=w1(λ0+λ1b11+⋯+λkb1k)+w2(λ0+λ1b21+⋯+λkb2k)+⋯+wn(λ0+λ1bn1+⋯+λkbnk)=λ0i=1∑nwi+λ1i=1∑nwibi1+⋯+λki=1∑nwibik=0

We have E(Ri)=λ0+λ1bi1+⋯+λkbikE(R_i)=\lambda_0+\lambda_1b_{i1}+\dots+\lambda_kb_{ik}E(Ri)=λ0+λ1bi1+⋯+λkbik satisfying ∑i=1nwiE(Ri)=wTμ=0\displaystyle \sum_{i=1}^{n}w_iE(R_i)=\pmb w^T\pmb \mu=0i=1∑nwiE(Ri)=wwwTμμμ=0. See λ0\lambda_0λ0 as risk-free return, i.e. λ0=Rf\lambda_0=R_fλ0=Rf, so we have
E(Ri)=Rf+λ1bi1+⋯+λkbikE(R_i)=R_f+\lambda_1b_{i1}+\dots+\lambda_kb_{ik} E(Ri)=Rf+λ1bi1+⋯+λkbik
Specifically, if one factor portfolio only depends on factor 1 (F1F_1F1), then
E(R1)=Rf+1⋅λ1⇒λ1=E(R1)−RfE(R_1)=R_f+1\cdot\lambda_1\Rightarrow \lambda_1=E(R_1)-R_f E(R1)=Rf+1⋅λ1⇒λ1=E(R1)−Rf
Similarly,
λ2=E(R2)−Rf…λk=E(Rk)−Rf\lambda_2=E(R_2)-R_f\\ \dots\\ \lambda_k=E(R_k)-R_f λ2=E(R2)−Rf…λk=E(Rk)−Rf
Therefore,
E(Ri)=Rf+bi1[E(R1)−Rf]+⋯+bik[E(Rk)−Rf]E(R_i)=R_f+b_{i1}[E(R_1)-R_f]+\dots+b_{ik}[E(R_k)-R_f] E(Ri)=Rf+bi1[E(R1)−Rf]+⋯+bik[E(Rk)−Rf]