投资学4-套利定价理论

2023-02-22 10:24发布

Investment 4-Arbitrage pricing theory1. The Arbitrage Pricing TheoryThe multi-fa
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1楼 · 2023-02-22 11:03.采纳回答

Investment 4-Arbitrage pricing theory

1. 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