Objective
Test whether the Bansal et al. log-linear approximation (Appendix B, Eqs. A.10-A.11) for zero-coupon equity strip pricing preserves enough accuracy for regime-switching models at horizons relevant to CRE pricing (20-50 years).
Setup
- Model: Bansal-Miller-Song-Yaron 2-regime long-run risk model at paper calibration
- Evaluator: log-linear recursion for strip prices z_{n,t}
- Comparators: MC simulation (5M effective paths, CRN) + exact grid recursion (deterministic, 200 x-points x 30 GH nodes)
- Acceptance: relative price error < 5% at 50yr in both regimes
Procedure
- Implement approximation recursion and MC benchmark independently
- Phase 3b: MC validation with 5 seeds x 1M paths, CRN-controlled
- Phase 4: head-to-head comparison at maturities 1yr-50yr in both regimes
- Phase 5: per-step Jensen gap decomposition and asymptotic decay analysis
- Phase 6: 13-perturbation sensitivity sweep across key parameters
- Audit: 3 independent methods (auditor MC, referee MC, exact grid) + adversarial fragility
Results
- Relative price error at 50yr: 59.8% (expansion), 66.9-72.2% (recession, MC/exact-grid)
- Slope overstatement (5y-1y unconditional): 12x
- Asymptotic decay distortion: 20.1-27.1%
- Per-step Jensen gap: 0.000576 (expansion), 0.00489 (recession) — recession 8.5x larger
- Short-horizon (1-3yr) errors below 1% — paper’s empirical results unaffected
- Sensitivity: all 13 perturbations produce positive error at 5yr+; worst driver is p_2=0.995 (+654%)
- Audit: all 5 sub-claims PASS; exact grid confirms MC to within noise; 168/168 tests pass
Analysis
The error is a deterministic Jensen gap, not MC noise: log(p_1 exp(a) + p_2 exp(b)) is not p_1 a + p_2 b, and this gap accumulates linearly with maturity. The recession regime dominates (larger risk price lambda and higher consumption volatility). The approximation decays too fast (asymptotic rate -0.052 vs -0.041 to -0.043 under MC/exact grid). At horizons where the paper has empirical data (1-3 years), errors are negligible; the concern is for the population term structure of equity risk premia computed from this recursion, which the paper uses as its headline structural claim.
Claim updates
- sdf-multiplicative-factorization-decomposes-long-run: The log-linear approximation of the multiplicative factorization introduces a deterministic Jensen gap that grows with maturity; the asymptotic decay rate under the approximation is distorted 20-27% from the true rate, affecting the connection between principal eigenvalue theory and the approximated strip prices.
- principal-eigenvalue-determines-long-run-asset: Empirically consistent — strip decay converges to a single regime-averaged rate — but the log-linear approximation overstates the decay rate by 20-27%, so the principal eigenvalue connection is quantitatively distorted at long horizons.
Follow-up
- Quantitative impact on Figure 2 / Tables 3-4 of the paper requires re-solving with exact pricing
- Higher-order approximation (quadratic in x) could reduce the gap but adds complexity
- JFE published version (Epstein-Zin SDF) likely has same Jensen vulnerability