Quantitative Models and Construction Methods Factor Covariance Matrix and Specific Volatility - **Model Name**: Factor Covariance Matrix - **Construction Idea**: The factor covariance matrix is used to capture the dynamic co-variation relationships between factors, providing a systematic framework for understanding market risk transmission mechanisms[3][18] - **Construction Process**: 1. **EM Algorithm**: Used to fill missing values in factor returns. The E-step estimates the conditional expectation of missing values, while the M-step re-estimates parameters iteratively until convergence Formula: $E[f_{mis}|f_{obs}]=\mu_{mis}+\Sigma_{mis,obs}\Sigma_{obs,obs}^{-1}(f_{obs}-\mu_{obs})$[21] Log-likelihood function: $L(\mu,\Sigma)=-\frac{T}{2}\big(D ln(2\pi)+\ln\big(\operatorname*{det}(\Sigma)\big)\big)-\frac{1}{2}\sum_{t=1}^{T}(f_{t}-\mu)^{\prime}\Sigma^{-1}(f_{t}-\mu)$[22] 2. **Half-life Weighted Adjustment**: Assigns exponentially decaying weights to historical data, emphasizing recent data[26] 3. **Newey-West Adjustment**: Corrects for heteroskedasticity and autocorrelation in time series data Formula: $\Sigma_{NW}=\Sigma_{0}+\sum_{i=1}^{L}w_{i}(\Sigma_{i}+\Sigma_{i}^{\prime})$[28] 4. **Eigenfactor Adjustment**: Addresses systematic underestimation of low-risk factor combinations using Monte Carlo simulations[35][38] 5. **Volatility Regime Adjustment (VRA)**: Adjusts factor volatilities to account for cross-sectional biases Formula: $\lambda_{F}=\sqrt{\sum_{t}(B_{t}^{F})^{2}w_{t}}$ $\tilde{\sigma}_{k}=\lambda_{F}\sigma_{k}$[53][54] - **Evaluation**: The factor covariance matrix effectively captures market co-variation relationships and provides reliable inputs for portfolio optimization[18][85] - **Model Name**: Specific Volatility - **Construction Idea**: Specific volatility focuses on predicting idiosyncratic risks at the stock level, addressing missing values and data anomalies[60] - **Construction Process**: 1. **Half-life Weighted Adjustment and Newey-West Adjustment**: Similar to the factor covariance matrix, but with different half-life settings for covariance and autocovariance matrices[61] 2. **Structured Model**: Adjusts for missing and anomalous data based on the relationship between specific volatility and factor exposures Formula: $\ln(\sigma_{n}^{TS})=\sum_{k}x_{nk}b_{k}+\epsilon_{n}$[67] 3. **Bayesian Shrinkage**: Reduces mean-reversion bias by shrinking estimates toward group averages Formula: $\sigma_{n}^{SH}=v_{n}\bar{\sigma}(g_{n})+(1-v_{n})\hat{\sigma}_{n}$[72] 4. **Volatility Regime Adjustment (VRA)**: Similar to factor volatility adjustment, but incorporates market-cap-weighted cross-sectional biases Formula: $\lambda_{S}=\sqrt{\sum_{t}(B_{t}^{S})^{2}w_{t}}$ $\tilde{\sigma}_{n}=\lambda_{S}\sigma_{n}^{SH}$[79][80] - **Evaluation**: Specific volatility adjustments improve the accuracy of idiosyncratic risk predictions, particularly for stocks with high data quality[60][73] --- Model Backtesting Results Factor Covariance Matrix - **Bias Statistic**: - Random portfolios: 1.05-1.06 - CSI 300: 1.15-1.19 - CSI 1000: 1.10-1.16[91] - **Q Statistic**: - Random portfolios: 2.73 - CSI 300: 2.95-2.97 - CSI 1000: 2.72-2.83[91] Specific Volatility - **Bias Statistic**: - Random portfolios: 1.06-1.07 - CSI 300: 1.19 - CSI 1000: 1.10[93] - **Q Statistic**: - Random portfolios: 2.73 - CSI 300: 2.97 - CSI 1000: 2.72[93] --- Quantitative Factors and Construction Methods Composite Fundamental-Price Factor - **Factor Name**: Composite Fundamental-Price Factor - **Construction Idea**: Combines low-frequency and high-frequency price-volume factors with fundamental factors to predict stock returns[128] - **Construction Process**: 1. **Lasso Model**: Uses a penalty coefficient of 0.001 to select features and predict market-neutralized stock returns[128] 2. **Factor Evaluation**: - RankIC: 7.43% - ICIR: 0.72 - Annualized long-short excess return: 61.15%[131] - **Evaluation**: The factor demonstrates strong predictive power but exhibits periodic underperformance during unfavorable market conditions[130] --- Factor Backtesting Results Composite Fundamental-Price Factor - **RankIC**: 7.43% - **ICIR**: 0.72 - **Annualized Long-Short Excess Return**: 61.15% - **Annualized Long-Only Excess Return**: 18.74%[131] 800 Index Enhancement Strategy - **Annualized Returns**: - Portfolio 1 (only stock deviation control): 18.28% - Portfolio 2 (stock/industry/style deviation control): 16.26% - Portfolio 3 (stock deviation + tracking error control): 17.81%[135][144] - **Tracking Error**: - Portfolio 1: 9.14% - Portfolio 2: 4.73% - Portfolio 3: 4.99%[135] --- Evaluation and Insights - The factor covariance matrix and specific volatility models provide robust risk predictions, enabling effective portfolio optimization and risk decomposition[85][152] - The composite fundamental-price factor demonstrates strong predictive ability but requires careful management of style and industry constraints to maintain alpha generation[130][136]