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Copula Black-Litterman Model with Machine Learning Derived Views and Uncertainty

Files

  • run_bt.py : Backtest all stategies.
  • copML_BL.py : Main framework for calculating weights.
  • copula_utils.py : Caculate copula covariance matrix, and functions for reading pre-trained data.
  • ml_utils.py : Machine learning models and feature generation.
  • optimize.py : Different objective functions for optimizing portfolio.
  • finance_data_util.py : Tools for fetching financial data and functions for calculating performance of portfolios.
  • ./data : Fundemental and macro datas.
  • ./fit_results : Pre-fit datas for copula and machine learning predictions.
  • ./results : Results for different strategies.

Overview

Enhanced the Black-Litterman model by incorporating vine-copula models for market equilibrium returns and ensemble machine learning for forecasting asset returns. Used ML model errors to quantify view uncertainty, improving portfolio performance and max drawdown in Taiwan’s stock market.

Data

  • Time Period: 2016.01.01 - 2024.03.01
  • Stocks: Top 50 market cap stocks in the US and Taiwan stock market.
  • Rebalance Frequency: 1 month
  • Data Sorce: Yahoo Finance

Methodology

Black-Litterman model

  • Black-Litterman model is consist of three parts:
1. Market Equalibrium
2. Personal View
3. Optimization
  • We use different models in each part of the optimization process to try to enhance the performance.

Vine-Copula Models

  • Captures the dependencies between stocks more accurately.
  • We use R package 'VineCopula' to calculate the covariance matrix more efficiently.
  • To make this process more smoothly, we use 'rpy2' to run R in a python script.

Ensemble Learning Models

  • Random Forest and XGBoost are used to predict stocks' returns.
  • Predict with technical analysis indexs and Macro datas.

Optimization Objective Functions

  • Max Sharpe Ratio
  • Minimize CVaR

Portfolio Construction

  • We test different combinations with different models, shown as below:
Portfolio Name Market Equakibrim Views Simulate with Copula Model Machine Learning Uncertainty
BL-RF BL Model Random Forest N N
Rvine-RF R-vine copula Random Forest Y Y
Cvine-RF C-vine copula Random Forest Y Y
Dvine-RF D-vine copula Random Forest Y Y
BL-XGB BL Model Xgboost N N
Rvine-XGB R-vine copula Xgboost Y Y
Cvine-XGB C-vine copula Xgboost Y Y
Dvine-XGB D-vine copula Xgboost Y Y

Backtest Results-TW

Taiwan Market 24 Month rolling window

tw_bst

Taiwan Market 60 Month rolling window

tw_MLcomp

Taiwan Portfolio Performances

Strategy Annual Return Annual Volatility Sharpe Ratio Sortino Ratio Max Drawdown Daily VaR
Panel A: Max Sharpe ratio
BL-RF 11.51% 10.39% 1.02 1.32 -13.86% 0.99%
Rvine-RF 30.74% 19.85% 1.41 1.97 -24.25% 1.99%
BL-XGB 19.77% 12.14% 1.48 2.04 -16.79% 1.17%
Rvine-XGB 31.32% 19.91% 1.43 2.00 -24.23% 2.00%
Panel B: Min CVaR
BL-RF 24.85% 21.92% 1.08 1.28 -26.36% 1.40%
Rvine-RF 24.07% 21.08% 1.09 1.69 -28.87% 2.12%
BL-XGB 40.85% 23.64% 1.53 1.92 -24.33% 1.63%
Rvine-XGB 24.59% 21.49% 1.09 1.69 -30.14% 2.10%
Panel C: Copula Simulated Minimum CVaR
R-vine RF 25.56% 21.15% 1.14 1.77 -27.52% 2.09%
R-vine XGB 26.67% 21.34% 1.18 1.84 -29.49% 2.05%
Panel D: Machine Learning Uncertainty
R-vine RF 32.39% 22.43% 1.33 1.84 -24.38% 2.32%
R-vine XGB 34.39% 21.97% 1.42 1.98 -24.65% 2.24%
Panel E: Hybrid Model Portfolio
R-vine RF 30.80% 22.31% 1.28 1.77 -25.65% 2.22%
R-vine XGB 36.40% 21.90% 1.49 2.09 -25.16% 2.28%
Panel F: Benchmarks
market weight 13.69% 18.94% 0.73 1.05 -27.40% 1.86%
equal weight 16.79% 16.58% 0.97 1.18 -28.07% 1.57%
max Sharpe 16.63% 11.43% 1.33 1.73 -16.80% 1.06%
min CVaR 10.48% 9.46% 1.01 1.32 -13.39% 0.90%

Backtest Result-US

Us Market 24 Month rolling window

us_bst

Taiwan Market 60 Month rolling window

us_MLcomp

Strategy Annual Return Annual Volatility Sharpe Ratio Sortino Ratio Max Drawdown Daily VaR
Panel A: Max Sharpe ratio
BL-RF 7.51% 16.88% 0.38 0.44 -28.63% 1.46%
Rvine-RF 14.69% 23.10% 0.61 0.77 -31.09% 2.17%
BL-XGB 10.26% 17.98% 0.50 0.60 -30.59% 1.60%
Rvine-XGB 16.79% 23.17% 0.69 0.87 -33.36% 2.19%
Panel B: Min CVaR
BL-RF 11.35% 18.14% 0.56 0.63 -30.22% 1.64%
Rvine-RF 18.47% 23.19% 0.75 0.96 -31.46% 2.20%
BL-XGB 10.76% 21.23% 0.48 0.53 -36.80% 1.81%
Rvine-XGB 20.20% 23.46% 0.80 1.03 -32.10% 2.22%
Panel C: Copula Simulated Minimum CVaR
R-vine RF 18.46% 23.41% 0.74 0.96 -31.04% 2.17%
R-vine XGB 21.06% 23.58% 0.83 1.05 -33.03% 2.23%
Panel D: Machine Learning Uncertainty
R-vine RF 17.30% 24.15% 0.69 0.88 -33.78% 2.32%
R-vine XGB 16.29% 24.18% 0.65 0.84 -34.76% 2.31%
Panel E: Hybrid Model
R-vine RF 18.41% 24.03% 0.73 0.93 -33.19% 2.28%
R-vine XGB 15.81% 24.03% 0.63 0.82 -33.74% 2.26%
Panel F: Benchmarks
market weight 10.29% 23.40% 0.44 0.55 -33.40% 2.22%
equal weight 12.90% 21.66% 0.56 0.69 -32.57% 1.86%
max Sharpe 6.38% 17.07% 0.31 0.37 -29.90% 1.54%
min CVaR 5.68% 16.73% 0.27 0.32 -29.26% 1.46%