Quantitative Models and Construction Methods 1. Model Name: Basic Risk Parity Model - **Model Construction Idea**: The model ensures that each asset in the portfolio contributes equally to the overall portfolio risk[20][23] - **Model Construction Process**: - Let the return vector of assets at time T be **r** and the weight vector be **w** - Covariance between assets is denoted as **Σ**, and the portfolio's return and volatility are: $$ \sigma(w) = \sqrt{w^T \Sigma w} $$ - Marginal Risk Contribution (MRC) and Risk Contribution (RC) for asset i are: $$ MRC_i = \frac{\partial \sigma(w)}{\partial w_i} = \frac{(\Sigma w)_i}{\sqrt{w^T \Sigma w}} $$ $$ RC_i = w_i \cdot MRC_i = w_i \cdot \frac{(\Sigma w)_i}{\sqrt{w^T \Sigma w}} $$ - Total Risk Contribution (TRC) is: $$ TRC = \sum RC_i = \sqrt{w^T \Sigma w} $$ - Risk parity requires: $$ RC_i = RC_j \; \text{for all} \; i, j $$[23][24][25] - **Model Evaluation**: The model is effective in balancing risk contributions but may lead to conservative portfolios when asset volatilities differ significantly[5][20] 2. Model Name: Risk Budgeting Risk Parity - **Model Construction Idea**: Adjusts the risk budget to allocate higher weights to riskier assets, making the model more flexible for different risk preferences[5][33] - **Model Construction Process**: - Adjust the relative marginal contribution of assets to the benchmark: $$ RC_i : RC = k_i \; \text{for all} \; i $$ - When assets are uncorrelated, the allocation becomes: $$ RC_i = \frac{k_i w_i^2 \sigma_i^2}{\sum k_i} $$ - Risk budget and actual weights are related quadratically: $$ \text{If actual weight is } n \times \text{basic weight, then risk budget is } n^2 $$ - Static and dynamic risk budgeting rules: - Static: Fixed risk budgets for equities, commodities, and gold - Dynamic: Adjust risk budgets based on Sharpe ratios over the past 6 months[37][39][41] - **Model Evaluation**: Provides higher returns but increases risk. Dynamic budgeting improves returns further but has mixed effects on risk metrics[41] 3. Model Name: Macro Risk Parity Model - **Model Construction Idea**: Allocates risk based on shared macroeconomic factors rather than individual asset risks, addressing overlapping risk contributions among assets[10][64] - **Model Construction Process**: - General asset pricing model: $$ r = 1^T \times I \times f_{base} + B \times I \times F + \varepsilon $$ - **I**: Dummy variable matrix indicating asset categories - **f_base**: Benchmark returns for major asset classes - **F**: Factor returns explaining intra-class differences - **B**: Factor exposures (sensitivity of assets to factors) - **ε**: Residual returns not explained by factors[64][66][68] - Systematic and idiosyncratic risk contributions: $$ RCF_i = w_{new,i} \cdot \frac{(\Sigma_f w_{new})_i}{\sqrt{w^T \Sigma w}} $$ $$ RCE_i = w_{new,i} \cdot \frac{(E w_{new})_i}{\sqrt{w^T \Sigma w}} = \frac{w_{new,i}^2}{\sqrt{w^T \Sigma w}} $$[74][75] - **Model Evaluation**: Effective in reducing leverage and addressing overlapping risks but requires precise macro risk modeling[12][115] --- Model Backtest Results 1. Basic Risk Parity Model - **Annualized Return**: 5.03% - **Maximum Drawdown**: -5.10% - **Volatility**: 2.58% - **Sharpe Ratio**: 1.90 - **Monthly Win Rate**: 71.11% - **Monthly Profit-Loss Ratio**: 3.44[28] 2. Risk Budgeting Risk Parity - **Static Risk Budgeting**: - **Annualized Return**: 5.80% - **Maximum Drawdown**: -9.30% - **Volatility**: 5.80% - **Sharpe Ratio**: 0.97 - **Monthly Win Rate**: 58.89% - **Monthly Profit-Loss Ratio**: 2.05 - **Dynamic Risk Budgeting**: - **Annualized Return**: 6.98% - **Maximum Drawdown**: -12.38% - **Volatility**: 6.29% - **Sharpe Ratio**: 1.07 - **Monthly Win Rate**: 63.33% - **Monthly Profit-Loss Ratio**: 2.30[46] 3. Macro Risk Parity Model - **Basic Asset Classes**: - **Annualized Return**: 5.03% - **Maximum Drawdown**: -5.10% - **Volatility**: 2.58% - **Sharpe Ratio**: 1.90 - **Monthly Win Rate**: 71.11% - **Monthly Profit-Loss Ratio**: 3.44 - **Expanded Sub-Asset Classes**: - **Annualized Return**: 7.35% - **Maximum Drawdown**: -11.49% - **Volatility**: 6.63% - **Sharpe Ratio**: 1.07 - **Monthly Win Rate**: 63.33% - **Monthly Profit-Loss Ratio**: 2.29[89] 4. Refined Asset Pool - **Asset Risk Parity**: - **Annualized Return**: 6.63% - **Maximum Drawdown**: -2.84% - **Volatility**: 2.83% - **Sharpe Ratio**: 2.27 - **Monthly Win Rate**: 75.51% - **Monthly Profit-Loss Ratio**: 5.99 - **Macro Risk Parity**: - **Annualized Return**: 8.03% - **Maximum Drawdown**: -3.59% - **Volatility**: 3.79% - **Sharpe Ratio**: 2.04 - **Monthly Win Rate**: 72.45% - **Monthly Profit-Loss Ratio**: 4.32[110]