Lambda M1 vs M2: Comprehensive Comparison

Bantleon's Risk Premium Calibration Methods | Realistic Historical Data (2600 days, 2017-11 to 2024-12)

1. Mathematical Framework

Method 1: Market-Based (Forward-Looking)

\(\lambda_{M1} = \frac{\ln\left(\frac{1 + y_{10Y}}{1 + r_f}\right)}{\beta_{ref}}\)
  • y10Y: 10-year government bond yield
  • rf: Risk-free rate (3-month)
  • betaref: Reference duration beta (=7.0)

Method 2: Historical (Backward-Looking)

\(\lambda_{M2} = \sum_{t=1}^{T} w_t \cdot f_t\)
  • ft: Factor return at time t
  • wt: Exponential decay weight
  • halflife: 24 months
Hybrid Blending Formula
\(\mu_{hybrid} = R^2 \cdot \mu_{M1} + (1 - R^2) \cdot \mu_{M2}\)

R-squared from factor regression determines the blend weight

2. Time Series Comparison

Lambda Comparison Over Time
Lambda Spread (M1 - M2)

3. Input Data

Yield Curve Inputs (M1)
Factor Returns (M2)

4. Sensitivity Analysis

M1 Sensitivity: beta_ref vs Yield Spread
M2 Sensitivity: Halflife vs Window

5. Regime Detection

R-squared Regime Indicator

6. Implied Returns Comparison

Implied Excess Returns by Method
M1 vs M2 Scatter

7. Summary Statistics

Metric Method 1 (Market) Method 2 (Historical) Comparison
Mean Lambda 0.093% -0.055% Difference: 0.148%
Std Dev Lambda 0.049% 1.022% M1/M2 Ratio: 0.05x
Correlation -0.066 Low agreement
Mean R-squared nan M2 preferred
Current Lambda (Final) 0.084% -0.013% Hybrid: nan%
Technical Details & Dependencies

Python Packages

Package Version Purpose
numpy 1.26.4 Array operations
pandas 2.3.3 DataFrame operations
plotly 5.24.1 Interactive charts
Data Source: Synthetic simulation
Factor Returns: Simulated bond duration proxy
Python Version: 3.12.3

Generated: 2026-01-17 09:13 | BRISMA Implied Risk Premia Framework | GitHub