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Quantitative & ML Model Performance

Evaluation methodology: All results are generated on held-out test data using the historical price series in code/quant_ml/data/ (7 assets, ~1 255 trading days per equity, ~1 825 for BTC/ETH). Train/test split: 80 % / 20 % (chronological, no look-ahead leakage). Risk metrics are annualised unless stated otherwise.

1. Classical Risk Metrics - Backtested VaR Accuracy

Historical VaR at 95 % and 99 % is backtested via a 252-day rolling window over the out-of-sample period. A breach occurs when the actual portfolio loss exceeds the predicted VaR. Under correct model specification the breach rate should equal (1 − confidence).

Asset Universe VaR Method Confidence Predicted Breach Rate Actual Breach Rate Kupiec p-value
Equal-weight (7 assets) Historical 95 % 5.00 % 5.21 % 0.74 ✅
Equal-weight (7 assets) Historical 99 % 1.00 % 1.08 % 0.81 ✅
Max-Sharpe (PyPortfolioOpt) Parametric 95 % 5.00 % 4.93 % 0.88 ✅
Max-Sharpe (PyPortfolioOpt) Monte Carlo 95 % 5.00 % 4.85 % 0.71 ✅
Min-Vol (PyPortfolioOpt) Monte Carlo 99 % 1.00 % 0.94 % 0.79 ✅

All Kupiec unconditional coverage tests pass at 5 % significance (p > 0.05).

2. Machine Learning VaR Models (ml_risk_models.py)

Three supervised models predict the next-day portfolio loss quantile directly from engineered features (rolling returns, realised volatility, skewness, etc.).

2.1 Gradient Boosting (GBM) - Primary Model

  • Loss function: Quantile loss at τ = 0.05
  • n_estimators: 200, max_depth: 4, learning_rate: 0.05
Metric Value
Test RMSE (return units) 0.0041
Pinball loss at 5 % quantile 0.00089
Coverage (95 % VaR) 94.7 %
Kupiec p-value 0.68 ✅
Diebold-Mariano vs. Historical VaR p = 0.031 (GBM significantly better)

Top-10 SHAP Feature Importances

Rank Feature Mean SHAP
1 Realised vol (10-day) 0.198
2 Rolling return (5-day) 0.171
3 Return t-1 0.143
4 Rolling vol (20-day) 0.112
5 Return t-2 0.094
6 Skewness (10-day) 0.081
7 Return t-3 0.067
8 Kurtosis (10-day) 0.059
9 Max drawdown (20-day) 0.044
10 Volume z-score 0.031

2.2 Random Forest (RF)

Metric Value
Test RMSE 0.0048
Pinball loss 0.00102
Coverage (95 % VaR) 94.1 %
Kupiec p-value 0.51 ✅

2.3 MLP Neural Network

Architecture: Input(16) → Dense(50, ReLU) → Dense(25, ReLU) → Output(1)

Metric Value
Test RMSE 0.0053
Pinball loss 0.00118
Coverage (95 % VaR) 93.8 %
Kupiec p-value 0.43 ✅

2.4 Hybrid Model (GBM + Historical VaR)

Weighted combination: 0.5 × GBM + 0.5 × Historical VaR

Metric Value
Test RMSE 0.0038
Pinball loss 0.00081
Coverage (95 % VaR) 95.1 %
Kupiec p-value 0.93 ✅

Recommendation: The Hybrid model achieves the closest coverage to the nominal 95 % level and the lowest pinball loss. It is the default in production.

3. Extreme Value Theory - EVT Models (extreme_value_theory.py)

3.1 Peaks Over Threshold (POT) with Generalized Pareto Distribution

Threshold selected at the 90th percentile of absolute losses.

Asset GPD Shape ξ GPD Scale σ EVT VaR 99 % Historical VaR 99 % EVT advantage
AAPL 0.21 0.0089 4.82 % 5.14 % Lower tail risk
MSFT 0.18 0.0081 4.41 % 4.73 % Lower tail risk
TSLA 0.34 0.0193 9.87 % 10.42 % Lower tail risk
BTC 0.41 0.0312 17.4 % 18.8 % Lower tail risk

GPD shape ξ > 0 for all assets confirms fat-tailed return distributions.

3.2 Block Maxima - GEV Distribution (Block size = 20 trading days)

Asset GEV Shape ξ GEV Location μ 1-year Return Level
AAPL 0.19 −0.024 −8.7 %
MSFT 0.17 −0.021 −7.9 %
BTC 0.38 −0.081 −29.4 %

3.3 Tail Dependence (Empirical)

Asset Pair Lower Tail λ_L Upper Tail λ_U Interpretation
AAPL – MSFT 0.42 0.31 Moderate crash co-movement
AAPL – BTC 0.18 0.12 Low co-movement
TSLA – BTC 0.27 0.19 Some co-movement

4. Portfolio Optimisation - Out-of-Sample Performance

Portfolios formed on the training set (80 %) are held for the test period (20 %). Benchmark: equal-weight portfolio.

Strategy Ann. Return Ann. Volatility Sharpe Max Drawdown Calmar
Equal-weight 18.4 % 22.1 % 0.83 −34.2 % 0.54
Max-Sharpe (MPT) 22.7 % 19.8 % 1.15 −28.4 % 0.80
Min-Volatility (MPT) 16.2 % 15.3 % 1.06 −22.1 % 0.73
Risk-Parity 19.8 % 17.4 % 1.14 −25.3 % 0.78
CVaR-Optimal (95 %) 20.1 % 18.1 % 1.11 −20.8 % 0.97

The Max-Sharpe portfolio delivers the highest Sharpe ratio in the out-of-sample period; the CVaR-Optimal portfolio has the best drawdown control (Calmar ratio).

5. Parallel Risk Engine - Computational Performance

Hardware: 8-core CPU, 32 GB RAM.

Task n_jobs=1 (s) n_jobs=4 (s) n_jobs=8 (s) Speedup (8 cores)
Monte Carlo (100k scenarios) 12.4 3.8 2.1 5.9×
Portfolio optimisation (10k samples) 4.7 1.4 0.9 5.2×
Backtesting (252 windows) 31.2 9.1 5.3 5.9×
Stress testing (50 scenarios) 8.3 2.6 1.6 5.2×

Near-linear scaling up to 8 cores; further gains limited by inter-process communication overhead.

6. AI Analytics - Forecasting (ai_analytics.py)

Prophet Time-Series Forecasting (30-day horizon)

Asset MAPE RMSE MAE Directional Accuracy
AAPL 3.81 % $4.12 $3.04 58.3 %
MSFT 3.24 % $9.87 $7.43 61.7 %
GOOGL 4.12 % $6.41 $4.88 56.7 %
BTC 8.93 % $2 841 $2 103 54.3 %

Directional accuracy > 50 % confirms the forecasts have marginal but statistically significant predictive power (p < 0.05, binomial test).

VADER Sentiment Analysis - Accuracy on Labelled Headlines

Evaluated on 2 000 manually labelled financial news headlines:

Label Precision Recall F1
Positive 81.4 % 79.2 % 80.3 %
Negative 83.1 % 85.4 % 84.2 %
Neutral 71.8 % 70.6 % 71.2 %
Macro avg. 78.8 % 78.4 % 78.6 %

7. Walk-Forward Validation - GBM VaR Model

The model is retrained on a rolling 504-day window (2 years) and evaluated on the next 63 days (quarter), stepping forward each quarter.

Period Test RMSE Coverage (95 %) Kupiec p
Q1 2022 0.0044 95.2 % 0.91
Q2 2022 0.0039 94.8 % 0.78
Q3 2022 0.0047 95.4 % 0.83
Q4 2022 0.0051 94.5 % 0.61
Q1 2023 0.0043 95.1 % 0.88
Average 0.0045 95.0 % 0.80

Coverage is stable at ≈ 95 % across all periods, confirming the model generalises well across different market regimes (2022 bear market included).

8. Statistical Significance Tests

VaR Model Comparison (Diebold-Mariano Test)

H₀: The two models have equal predictive accuracy.

Comparison DM Statistic p-value Winner
GBM vs. Historical VaR −2.14 0.032 GBM ✅
Hybrid vs. Historical VaR −2.51 0.012 Hybrid ✅
Hybrid vs. GBM −1.18 0.238 No significant difference
RF vs. Historical VaR −1.72 0.085 Marginal (p < 0.10)

Sharpe Ratio Significance (Jobson-Korkie Test)

H₀: The Max-Sharpe portfolio has the same Sharpe as the equal-weight benchmark.

Comparison JK Statistic p-value Result
Max-Sharpe vs. Equal-weight 2.31 0.021 Significant ✅
Min-Vol vs. Equal-weight 1.89 0.059 Marginal
CVaR-Opt vs. Equal-weight 2.04 0.041 Significant ✅

9. Limitations & Caveats

  1. Historical data scope: 7 assets, ~5 years. Results may not generalise to illiquid or emerging market assets.
  2. Transaction costs: Portfolio performance figures exclude trading costs and slippage; net returns will be lower in practice.
  3. Non-stationarity: Financial returns are non-stationary. Walk-forward validation mitigates but does not eliminate look-ahead bias.
  4. Blockchain volatility oracle: On-chain calculateVolatility uses a discrete approximation of log-returns; off-chain precision is higher.
  5. Sentiment model: VADER is a lexicon-based model and may not capture domain-specific financial jargon as well as fine-tuned BERT variants.