Objective

• Create a nonlinear version of IPCA using autoencoders

Results

• Reduces out-of-sample pricing errors (predictive \(R^2\))
• Imposes economic restriction of no-arbitrage (no intercept)

A model of returns

\[r_{i,t+1}=\alpha_{i,t}+\beta'_{i,t}f_{t+1}+\epsilon_{i,t+1}\]

• \(r_{i,t+1}\): return for stock \(i\) at time \(t+1\)
• \(f_{t+1}\): systemic risk factors at time \(t+1\)
• \(\beta'_{i,t}\): exposure of stock \(i\) at \(t+1\) to systemic risk factors
• \(\alpha_{i,t}\): intercept term (perhaps set to 0)
• \(\epsilon_{i,t+1}\): error

A model of returns

\[r_{i,t+1}=\alpha_{i,t}+\beta'_{i,t}f_{t+1}+\epsilon_{i,t+1}\]

• \(f_{t+1}\): systemic risk factors at time \(t+1\)
  • These are the "factors" in a factor model
  • Systemwide: no \(i\) subscript
  • Can be pre-specified or latent
  • We will use latent factors

IPCA

• From "Characteristics are covariances" by KPS
• Idea: characteristics proxy for exposure to risk factors
  • Momentum, volatility, bid-ask spread
• Conditional exposures: \(\beta(z_{i,t})'=z'_{i,t}\Gamma_\beta\)
• \(z_{i,t}\): vector of characteristics of asset \(i\)

IPCA

• Provides robust interpretation of returns
• If characteristics proxy for risk factors, then \(\beta\neq 0\) and \(\alpha=0\)
• If not, then characteristics can be used for compensation without risk, so \(\beta=0\) and \(\alpha\neq 0\)
  • This is an "anomaly" (arbitrage)

IPCA Analysis

• \(R^2_{total}\): fraction of variance in \(r_{t+1}\) explained by \(\hat{\beta}_{i,t}'\hat{f}_{t+1}\)
  • Ability of model to explained realized variation in returns (systemic risks)

IPCA Analysis

• \(R^2_{predictive}\): fraction of variance in \(r_{t+1}\) explained by \(\hat{\beta}_{i,t}'\hat{\lambda}\)
  • ​\(\hat\lambda\): vector of estimated risk factor prices
  • \(\hat{\beta}_{i,t}'\hat{\lambda}\): model-based conditional expected return on asset \(i\) given \(t\) information
  • Measures the accuracy of model-implied conditional expected returns 
    • Ability to describe differences in average returns (risk compensation).

IPCA Results

• Achieves similar \(R^2_{total}\) to Fama-French (in sample)
• Better \(R^2_{total}\) out of sample
• More than double Fama-French in \(R^2_{predicitve}\)
• \(\alpha\) usually insignificant from zero for 5-factor
  • ​If significant, returns are small

Conditional Autoencoder

• Covariates improve estimates of factor loadings and latent factors
• Design new neural network structure by augmenting a standard autoencoder to incorporate covariates
• \(r_{i,t}=\beta'_{i,t-1}f_t+u_{i,t}\)

Conditional Autoencoder

Beta (left side)

• Neural network to compute betas:

Factor (right side)

• Neural network to compute betas:

• \(L_f=1\)
  • This makes the factors interpretable as portfolios

Factor (right side)

• Difficult to use full cross section of individual stock returns 
  • Many weight parameters: 30,000 firms, 720 months
  • Panel is unbalanced: only 6,000 stocks/month
• Solution: initialize network with set of portfolios
  • \(x_t=(Z'_{t-1}Z_{t-1})^{-1}Z_{t-1}r_t\)
  • Set of portfolios dynamically re-weighted by characteristic
  • \(j^{th}\) element: return of long-short portfolio constructed by sorting stocks based on \(j^{th}\) characteristic

Objetive Function

• \(\theta\): summarizes weight parameters
• \(\phi(\theta)\): penalty function for regularization
  • Use LASSO (\(l_1\))
  • \(\phi(\theta;\lambda)=\lambda \sum_j |\theta_j|\)
  • Set coefficients on a subset of covariates to exactly zero
  • Imposes sparsity on weights

Other Regularization Techniques

• Early stopping: stop when validation sample errors begin to increase
  • Usually occurs before errors minimized in training
• Ensemble: train 10 networks and use average prediction

Optimization Algorithn

• Stochastic gradient descent
• Adam optimizer
• Batch normalized: for each hidden layer in each training step (batch), cross-sectionally de-mean and standardize
  • Motivated by internal covariate shift: inputs of hidden layers follow different distributions than their counterparts in the validation sample
  • Should restore representation power of the unit

Characteristics

• 94 characteristics
  • 61 updated annually
  • 13 updated quarterly
  • 20 updated monthly
• Delay characteristics to avoid forward looking bias:
  • Monthly by 1 month
  • Quarterly by 4 months
  • Annually by 6 months
• Avoid recursively refitting model each month
  • Refit annually (most signals annual)

Characteristics

• Missing characteristics replaced by cross-sectional median of that characteristic for that month
• Distributions can be skewed and leptokurtic
  • Rank-normalize characteristics
  • Create 94 managed portfolios
  • Also include one equal weighted market portfolio
• No filters based on prices or share types

OOS \(R^2_{total}\)

• IPCA with 6 factors top performing
• Closely followed by CA networks
• FF models worst:
  • Infrequent re-estimation of parameters
  • Much larger cross-section of stocks than normal

OOS \(R^2_{total}\)

OOS \(R^2_{total}\)

OOS \(R^2_{pred}\)

• CAs did much better than IPCA
• Static models did poorly

OOS \(R^2_{pred}\)

OOS \(R^2_{pred}\)

Economic Performance

• Sort stocks based on OOS forecast
• Create zero-net investment portfolio
  • Buy top 10%
  • Sell bottom 10%
• Equal-weighted and value-weighted portfolios
• CA2 top performing

Economic Performance

Economic Performance

• Constructed using mean and covariance matrix of estimated factors through \(t\) and tracking the post-formation \(t+1\) return
• CA3 5-factor top performing
• Not necessarily implementable strategies

Risk Premia vs. Mispricing

• Models above specified with no intercept (\(\alpha\))
  • Imposes no arbitrage
  • Stock characteristics proxy for compensated factor risk exposures
• Should there be an intercept?

Risk Premia vs. Mispricing

• If zero-intercept is correct model, time series average of model residuals for each asset should be statistically indistinguishable from zero
• \(\alpha_i := E[u_{i,t}]=E[r_{i,t}]-E[\beta'_{i,t-1}f_t]\)
• Use \(t\)-tests

Risk Premia vs. Mispricing

• Magnitude of alphas less for CA
• Fewer are significant
  • Those that are have small magnitude (7 bps/mo)

Characteristics Importance

• Variable importance by reduction in \(R^2_{total}\) when removed
• Top 20 characteristics contributed 90% for CA1-3
• Three influential categories:
  • Price trend: reversal, momentum
  • Liquidity: turnover, dollar volume, bid-ask spread
  • Risk measures: volatility, market beta