Statement
Long-lived asset prices in affine representative-agent economies with recursive utility and rare events can be computed exactly as integrals of equity strip prices, each strip being exponential-affine in the state variables with coefficients solving Riccati ODEs. This strip-integral method requires only a single log-linear approximation (for the wealth-consumption ratio) rather than the standard double approximation, and is fully exact in two special cases: EIS psi=1 and time-additive utility (theta=1).
Evidence summary
Tsai and Wachter (2018) derive the method in a general affine jump-diffusion framework:
- Each equity strip H(D,x,tau) = D * exp(a_phi(tau) + b_phi(tau)‘x) where the coefficient functions solve Riccati ODEs (Theorem 13).
- Total equity price G(x) = integral of exp(a_phi(tau) + b_phi(tau)‘x) dtau (Corollary 14).
- When psi=1: wealth-consumption ratio is constant (beta^{-1}), so the single approximation is exact.
- When theta=1 (time-additive): the SDF does not depend on the value function, so no approximation is needed.
- Quantitative evaluation: in a Wachter (2013) rare disaster model extended to non-unitary EIS, the strip method is notably closer to numerical solutions than double log-linearization.
Conditions and scope
- Requires the affine structure (33a-33e) in Tsai-Wachter for state dynamics.
- The integral must converge (related to the no-bubble condition; see no-bubble-condition).
- For general psi != 1 and theta != 1, the wealth-consumption ratio approximation introduces errors, but these are smaller than the double-linearization errors documented in higher-order-effects-asset-pricing-models.
Counter-evidence
None identified. The method is mathematically proven for the two exact cases and validated numerically for the general case.
Linked ideas
Open questions
- How does the strip-integral approach connect to the discrete-time coupled Riccati recursions used in regime-switching asset pricing?
- Can the convergence of the strip integral be used as a practical no-bubble condition diagnostic?