Investment 4-Arbitrage pricing theory1. The Arbitrage Pricing Theory

The multi-factor model

The idea is to encode information on N assets with K factors (K<N) such that the residuals(the part unexplained by the factors) are uncorrelated(i.e., idiosyncratic)

• K-factor model: excess returnsR_j^e=\alpha_j+\beta_{j1}F_1+\beta_{j2}F_2+...+\beta_{jK}F_K+\epsilon_j
• E[F_k]=0, E[\epsilon_j]=0,E[\epsilon_jF_k]=0,E[\epsilon_i\epsilon_j]=0(i\ne j)
• factors are innovations in portfolio returns,F_k=R_{F_k}-E[R_{F_k}]
• In matrix notation R^e=\alpha+\beta F+\epsilon
• \mu-R_0=\alpha
• \Sigma=\beta\Sigma_F\beta^\top+\Sigma_{\epsilon} (total risk=systematic factor risk+idiosyncratic risk)
• this is a K-factor model of the covariance structure of returns

Arbitrage Pricing Theory links K-factor model of covariance of returns to K-factor model of expected returns

• No arbitrage ⇒ there exist \lambda_1,...,\lambda_K such that \mu-R_0=\beta\lambda ⇒ \alpha in the factor model cannot be chosen independently of \beta
• the expected excess returns on factor portfoliosλ_k =E[R_{F_k}]−R_0
• APT impliesR^e = βR_F^e + ε ⇒ Expected excess returns are determined by stock-specific factor loadings \beta and common factor risk-premia λ

λ是每单位灵敏度的某因素的预期收益溢价，也就是风险溢价
factor loading \beta 是因素载荷矩阵，也就是敏感度系数矩阵

Maximun Sharpe Ratio Portfolio

Suppose the portfolio has the return R_i^e=\alpha_i+\beta_iR_F^e+\epsilon_i with \alpha_i>0 and Var(\epsilon_i)=\sigma_i^2

• r_i=R_i^e-\beta_iR_F^e=\alpha_i+\epsilon_i is a zero-cost portfolio with zero factor exposure

此时是系统风险，还没有加入非系统风险因子，需要找到投资组合方案中的权重 w_F,w_i （a为变动的、可获得不同资产组合风险的风险厌恶指数），使maxE[R_p]-\frac{a}{2}V[R_p] ，即 max_{X_F,X_i}R_0+x_F(\mu_F-R_0)+x_i\alpha_i-\frac{a}{2}(x_F^2\sigma^2_F+x_i^2\sigma^2_i)

• FOC: x_i=\frac{\alpha_i}{a\sigma_i^2},x_F=\frac{\mu_F-R_0}{a\sigma_F^2}
• so the portfolio with maximum Sharpe ratio has weights x_i in R_i , x_F-\beta_ix_i in R_F and 1-x_F-x_i(1-\beta_i) in R_0

Information Ratio

The portfolio max Sharpe ratio is SR_P=\frac{\mu_P-R_0}{\sigma_P}=\sqrt{SR_F^2+IR^2}

• the benchmark Sharpe ratio is SR_F=\frac{\mu_F-R_0}{\sigma_F}
• the new strategy's Information Ratio is IR=\frac{\alpha_i}{\sigma_i} , measures the increase in SR due to the new \alpha_i>0 strategy
• the total risk of the portfolio is \sigma_P=\frac{SR_P}{a}

信息比率表示单位主动风险所带来的超额收益，从主动管理的角度描述风险调整后收益，不同于Sharpe Ratio从绝对收益和总风险角度来描述
信息比率越大，说明基金经理单位跟踪误差所获得的超额收益越高，因此，信息比率较大的基金的表现要优于信息比率较低的基金

Extends to n strategies with all α_i\ne 0 and uncorrelated \epsilon_i :

• max SR portfolio invests x_i=\frac{\alpha_i}{a\sigma_i^2} in each strategy andSR_P=\sqrt{SR_F^2+\Sigma_{i=1}^{n}IR_i^2}
  
• total risk of the portfolio is \sigma_P=\frac{SR_P}{a} (consistent with APT asymptotic arbitrage lim_{n\rightarrow \infty}SR_p=\infty )
• this applies to a multi-factor benchmark model(SRF is then the Sharpe ratio obtained with the K benchmark factor portfolios)

2. APT vs. CAPM

CAPM

• Expected excess returns are proportional to market beta
• The Market is the only source of priced risk
• CAPM is an equilibrium model which restricts the cross-section of expected returns given some assumptions on investor preferences, but does not require that one knows the covariance structure of returns.

APT

• Expected return are generated by exposures (factor betas) to several sources of risk times factor risk-premia
• APT gives no specific guidance as to what the systematic risk sources are
• APT relies on an (approximate) arbitrage argument to restrict the cross-section of expected returns given that one knows the fundamental covariance structure of returns

APT and CAPM are both consistent if the market portfolio return is ‘spanned’ by the K factors

APT和CAPM的公式形式是一样的，但内在的经济含义不同，CAPM是在市场均衡的条件下得到的，而APT是在无套利条件下得到的
均衡的市场里一定没有套利机会，而无套利机会并不意味着是均衡市场

Factors for a good APT model

How many factors?

• There are more factors that explain the covariance structure of security returns, than there are factors explaining the cross-section of expected returns
• Some factors (e.g., industry) carry zero risk-premia (\lambda=0 )
• \epsilon_i -residuals are largely uncorrelated \rightarrow a good factor model of covariance structure
• A mean-variance efficient portfolio \rightarrow a good factor model for the cross-section of expected returns
• adding additional factors does not improve the mean-variance efficient frontier spanned by the K factors
• For the cross-section of expected returns the number of factors in the APT model is not relevant as one can always reduce the number of factors to K = 1 by using a mean-variance efficient portfolio return as the unique pricing factor

How to search factors?

• Factors as portfolio returns → Fama-French Model
• Factors as asset characteristics → Barra Risk Model
• Factor analysis → Purely statistical analysis (e.g. principal component)→ Not clear how to interpret the factors → Prone to data-mining
• Factors as macro-variables → Classic paper Chen, Roll, and Ross (1986) → Macro-variables observed at low frequency and noisy → Factor-mimicking portfolio

3. Factors as portfolio returns: The Fama-French model

Fama and French show that two firm characteristics, other than beta, predict returns:

• Size: market capitalization (or ME)
• ME=Market Equity=number of shares × share price
• small firms (low ME) historically outperformed large firms
• however, in the last 30 years the size anomaly has been considerably weaker
• size remains useful in multi-variate sorts (e.g., the value anomaly is particularly strong among small firms)
• Value: book-to-market ratio (or BM)
• BM=BE/ME, the ratio of a firm’s book value of equity to its market capitalization (same as book-to-price)
• value firms (high BM) consistently outperform growth (low BM) firms, even controlling for market beta

The Fama-French three-factor Model

• R_{j,t}-R_{0,t}=\alpha_j+\beta_{j,M}(R_{M,t}-R_{0,t})+\beta_{j,s}SMB_t+\beta_{j,h}HML_t+\epsilon_{j,t}
• SMB_t (Small minus Big) is the difference between the returns on diversified portfolios of small and big stocks
• HML_t (High minus Low) is the difference between the returns on diversified portfolios of high and low B/M stocks

HML和SMB的factor performance计算方法：先用OLS回归HML的表现，“beat the market”，即HML对 R_M^e=R_M-R_0 的回归结果，用最大Sharpe-Ratio法得到optimal portfolio；再加入SMB因子，用OLS回归出SMB关于 R_M^e 和HML的结果，运用maxSR得到新的最优资产组合

The Carhat Four-factor Model

• R_{j,t}-R_{0,t}=\alpha_j+\beta_{j,M}(R_{M,t}-R_{0,t})+\beta_{j,s}SMB_t+\beta_{j,h}HML_t+\beta_{j,u}UMD_t+\epsilon_{j,t}
• Momentum
• UMD_t (Up minus Down) is the difference between the returns on diversified portfolios of winner and loser stocks
• Rank each stock by its past return over the past year, typically exclude the most recent month in the year to avoid the short term reversal
• Create a portfolio that buys stocks that have gone Up (stocks in the top 3 deciles) and shorts stocks that have gone Down (stocks in the bottom 3 deciles)
• Hold this portfolio for the next month
• The return on this zero-cost long-short portfolio is UMD (‘up minus down’) or WML (winners minus losers’)
• The UMD factor has much higher turnover than HML and SMB

加入动量因子：对UMD因子进行OLS回归---UMD与market、size和value呈现负相关，可以用于对冲

Size&Value: size is useful characteristic when interacted with value in double-sorted and value-weighted portfolios & there is a value effect in each size group but much stronger among small cap stocks（小盘股）

Size&Momentum: the momentum effect is also present in each size group, but is stronger among small cap stocks

\rightarrow refine factors using double-sortedf portfolios: HML_s,UMD_s

4. Factors as asset characteristics

Use observable factor loading \beta_{jk} (asset characteristics such as market-cap, dividend-yield, industry and country dummy) and no need to estimate it

Run Fama-MacBeth cross-sectional regressions every month t R_{j,t}^e=\lambda_{0,t}+\sum_{k=1}^{K}\lambda_{k,t}\beta_{jk}+u_{j,t}, \ j=1,...,N

• use time-series of estimated \hat{\lambda}_{k,t} to estimate the factor risk-premium \bar{\lambda}_k=\frac{1}{T}\sum_{t=1}^{T}\hat{\lambda}_{k,t} and its standard error \sigma(\bar{\lambda}_k)=\frac{\sigma(\hat{\lambda}_{k})}{\sqrt{T}} where \sigma(\hat{\lambda}_k)^2=\frac{1}{T}\sum_{t=1}^{T}(\hat{\lambda}_{k,t}-\hat{\lambda}_{k})^2
• test whether intercept and factor risk premia are significant using a t-test applied to t-stat= \frac{\bar{\lambda}_k}{\sigma{(\bar{\lambda}_k)}}

Example: the Barra Europe Equity Model

Covers a universe of around 9500 assets and EUE3, the base version uses a total of 68 equity factors: a regional market factor, 29 country factors, 29 industry factors and 9 style factors (composites of 25 asset characteristics)

The style factors

• Size
• Log of the month-end issuer capitalization
• Log of total assets; an indicator of fundamental firm size
• Value
• Book-to-price ratio: the last published book value of common equity divided by the current issuer capitalization
• Sales-to-price ratio: sales over the last 12 months divided by the current issuer capitalization
• Momentum
• Historical weekly alpha. Intercept of a regression of weekly asset returns against weekly returns of the cap-weighted market portfolio. An exponentially-weighted average of 104 weeks of data is used
• 12-month relative strength (essentially a trailing excess return)
• 6-month relative strength (essentially a trailing excess return)
• Liquidity
• Log of annual share turnover; an indicator of the average liquidity of an asset over one year
• Log of quarterly share turnover
• Log of monthly share turnover
• Volatility
• Historical weekly beta
• Realized range of excess returns
• Realized return volatility
• Leverage
• Book leverage
• Market leverage
• Earnings Yield
• Trailing earnings-to-price ratio. Net earnings over the last 12 months divided by the current issuer capitalization
• Cash earnings-to-price ratio. Cash earnings over the last 12 months divided by the current issuer capitalization
• Return on equity. Net earnings over the last 12 months divided by the last available book value of common equity
• Predicted earnings-to-price ratio using 12-month forward-looking earnings per share
• Dividend Yield
• Annualized dividend per share divided by the current price
• Growth
• Trailing growth of total assets
• Trailing growth of annual sales
• Trailing growth of annual net earnings
• Short-term predicted earnings growth
• Long-term (3-5 years) predicted earnings growth

5. Aymptotic Aribtrage 渐近套利

APT formula without residual risk

Suppose K=1 and \epsilon_i=0 , so that ∀i R_i^e =α_i +β_iF

• Choose w_i,w_j such thatw_iβ_i +w_jβ_j =0 (⋆) andR_p^e =w_iR_i^e +w_jR_j^e =w_iα_i +w_jα_j +(w_iβ_i +w_jβ_j)F =w_iα_i +w_jα_j and no arbitrage implies w_iα_i +w_jα_j =0 , so \frac{\alpha_i}{\beta_i}=\frac{\alpha_j}{\beta_j}=\lambda
• \rightarrow No arbitrage implies the APT: R_i^e =β_i(F+λ)
• For a ‘traded factor’ F=R_F −E[R_F] and since β_F =1 its excess return satisifies R_F^e = (F + λ) so the APT becomes: R_ i^ e = β_ i R_ F^e
• If \epsilon_i\ne 0, then any finite portfolio which satisies (⋆) will have residual risk and thus the arbitrage argument will not hold for any finite economy → Instead, APT is justified by the absence of aymptotic arbitrage

APT with residual risk: Asymptotic Arbitrage

Suppose K=1 and \epsilon_i\ne 0 and that the common factor is and excess return, so that ∀i R_i^e =α_i +β_iR_F^e +\epsilon_i

• Consider the zero-investment portfolio with return r_i=R_i^e-\beta_iR_F^e=\alpha_i+\epsilon_i andE[r_i]=\alpha_i , V[r_i]=\sigma_{\epsilon_i}^2
• so R_P=\frac{1}{N}\sum_{i=1}^Nr_i , E[R_P]=\frac{1}{N}\sum_{i=1}^N|\alpha_i| andV[R_P]=\frac{1}{N^2}\sum_{i=1}^N\sigma_{\epsilon_i}^2\leq \frac{1}{N}max_i\sigma_{\epsilon_i}^2
• the maximum residual's variance is finite then lim_{N\rightarrow\infty}V[R_p]=0
• to rule out an arbitrage we must have lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^N|\alpha_i|=0
• this implies that all but a finite number of securities must have αi = 0, i.e., the APT must hold for most securities

Theorem

An asymptotic arbitrage is a sequence of portfolio weight vectors w^n such that

• 1^{\top}w^n=0
• lim_{n\rightarrow\infty}w^{n\top}\Sigma w^n=0
• w^{n\top}\mu\geq \delta>0

Suppose V[\epsilon_i]=S_i^2<\bar{S}^2 \ \forall i then No Asymptotic Arbitrage implies \mu=\lambda_01+\beta\lambda+\nu (⋆) where the vector of pricing errors \nu satisfies

• \nu^{\top}1=0
• \beta^{\top}\nu=0
• lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n\nu^2_i=0

If there is no asymptotic arbitrage, in the limit as the number of assets becomes large, most (that is, all but a finite number) of assets’ expected returns should satisfy the APT relation (⋆)

However, APT holds except that fully diversified portfolios with well-balanced weights (i.e., lim_{n\rightarrow\infty}\sum_{i=1}^n\frac{w_i^2}{1/n}\leq C for some constant C) have no idiosyncratic risk in the limit (i.e., lim_{n\rightarrow\infty}\sum_{i=1}^n{w_i^2}S_i^2=0 ) and lim_{n\rightarrow\infty}\sum_{i=1}^n{w_i}\nu_i=0 , so the portfolio \mu_P=\lambda_0+w^{\top}\beta\lambda

Equal weighted vs. value weighted portfolio returns

Suppose two stocks have the price path A=(100,200,100) and B=(100,100,100)

• EW portfolio return 0.5x(1−0.5)=25%(A有价格波动，100到200收益率是1，200到100收益率是-0.5，A的权重是0.5)
• VW portfolio return 1x1/2−0.5x2/3=17%(A从100到200收益率是1， 权重是1/2，从200到100时权重发生变化，为2/3)

VW portfolio return is a buy and hold strategy that requires no-trading

EW portfolio return is an implicit dynamic trading strategy, requires selling stocks that went up and buying stocks that went down

EW portfolio returns can be misleading:

• Stocks experience negative autocorrelation as a result of bid-ask bounce (not a ‘true’ return)
• It puts same weight on small illiquid stocks and on large liquid stock returns (loads on size and illiquidity anomalies)
• It loads on the short-term reversal anomaly (can be tested by increasing the return horizon)

Constructing covariance matrices

• Covariance matrices require lots of parameters: with 100 assets, an unconstrained covariance matrix has 5050 (N(N + 1)/2) parameters
• Parsimonious covariance matrices can be constructed with factor models. Assuming K independent factors with factor variance
  Cov(R_i,R_j)=\sum^K_{k=1}β_{jk}β_{ik}σ^2_k and Var(Ri)=\sum^K_{k=1}β^2_{ik}σ^2_k+ σ^2_{\epsilon_i} will require only (N+1)xK parameters for the covariances and N idiosyncratic volatilities. So for 100 assets and K=5 factors that’s 605 parameters
• Given a history of factor returns, one can construct covariance matrix forecasts by forecasting factor variances