[huuan26]

Summary

Expands the benchmark-relative performance analytics in app/services/backtest/metrics.py and Strategy API V2 with Modigliani–Modigliani ($M^2$), Dual-Beta ($\beta^+$ and $\beta^-$), and Batting Average metrics:

1. Modigliani–Modigliani Measure ($M^2$) & $M^2 \text{ Alpha}$:
   • Leverages or deleverages the portfolio return to match the benchmark's total volatility level ($\sigma_{p^*} = \sigma_b$), converting the abstract Sharpe ratio into an annualized percentage return directly comparable to the benchmark:
     $$M^2 = R_f + \text{Sharpe}p \times \sigma{b,\text{ann}} = R_f + \left(\frac{R_{p,\text{ann}} - R_f}{\sigma_{p,\text{ann}}}\right) \sigma_{b,\text{ann}}$$
     $$M^2_{\text{alpha}} = M^2 - R_{b,\text{ann}}$$
   • Directly answers: "If this strategy were scaled to carry the exact risk of the benchmark (e.g. SPY or BTC), what annualized return and excess alpha would it produce?"
   • Safely handles flat portfolio variance (zero_portfolio_variance) and flat benchmark variance (zero_benchmark_variance).
2. Dual-Beta: Up-Market ($\beta^+$) & Down-Market ($\beta^-$) Beta:
   • Asymmetric systematic risk modeling across distinct market regimes:
     $$\beta^+ = \frac{\text{Cov}(R_p, R_b \mid R_b > 0)}{\text{Var}(R_b \mid R_b > 0)}$$
     $$\beta^- = \frac{\text{Cov}(R_p, R_b \mid R_b < 0)}{\text{Var}(R_b \mid R_b < 0)}$$
   • Computes betaSpread = \beta^+ - \beta^-. A positive spread represents a convex payoff structure with high upside participation and defensive downside insulation.
   • Pairs directly with the Up/Down Capture Ratios added in feat(backtest): expand benchmark-relative metrics with Beta, Jensen's Alpha, Treynor, and Capture Ratios #303.
3. Batting Average & Outperformance Payoff Ratio:
   • battingAverage: Percentage of aligned observation periods where portfolio return strictly outperformed benchmark return:
     $$\text{Batting Average} = \frac{\sum_{t=1}^T \mathbb{I}(R_{p,t} > R_{b,t})}{T}$$
   • outperformancePayoffRatio: Ratio of average active return during winning periods to average active loss during losing periods.
   • Also records outperformancePeriods and underperformancePeriods.
4. Comprehensive Integration & Full Backward Compatibility:
   • Seamlessly integrated into calculate_benchmark_relative_metrics, populating mSquaredStatus, dualBetaStatus, battingStatus, and associated fields in result["benchmarkRelativeMetrics"].
   • Standalone functions exposed in __all__: calculate_m_squared, calculate_dual_beta, and calculate_batting_average.
   • 100% backward-compatible with existing callers.
5. Documentation:
   • Updated docs/product/BACKTEST_CENTER.md and docs/product/BACKTEST_CENTER_CN.md with mathematical formulas and explanations.

Related issue

• Follow-up to feat(backtest): expand benchmark-relative metrics with Beta, Jensen's Alpha, Treynor, and Capture Ratios #303
• Relates to Proposal: benchmark-relative Information Ratio for fixed-income fund analytics #179 (benchmark-relative fund analytics)

Changes

• backend_api_python/app/services/backtest/metrics.py:
  • Implement _calculate_m_squared_metrics, _calculate_dual_beta_metrics, and _calculate_batting_metrics.
  • Implement calculate_m_squared, calculate_dual_beta, and calculate_batting_average.
  • Wire new metrics into calculate_benchmark_relative_metrics.
  • Export new functions in __all__.
• backend_api_python/tests/test_backtest_metrics.py:
  • Add unit tests for $M^2$ (known values, positive risk-free rate, zero portfolio/benchmark variance, input validation).
  • Add unit tests for Dual-Beta (known values, upside/downside regime filtering, single-regime curves).
  • Add unit tests for Batting Average and Outperformance Payoff Ratio (known values, all-win edge case).
  • Add integration test for calculate_benchmark_relative_metrics.
• docs/product/BACKTEST_CENTER.md & BACKTEST_CENTER_CN.md:
  • Document $M^2$, Dual-Beta, and Batting Average in English and Chinese.

Test plan

• pytest backend_api_python/tests/test_backtest_metrics.py passes (31/31 tests)
• pytest backend_api_python/tests/test_strategy_v2_service.py passes (30/30 tests)
• ruff check app/services/backtest/metrics.py tests/test_backtest_metrics.py passes with zero violations
• python backend_api_python/scripts/backend_quality_check.py passes
• git diff --check passes with zero whitespace or line-ending issues