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Pricing Long-Lived Securities in Dynamic Endowment Economies

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Problem

Representative-agent asset pricing models with recursive utility and rare events are difficult to solve exactly. The standard Campbell-Shiller log-linearization is applied twice (once for the wealth-consumption ratio, once for equity prices), introducing compounding approximation errors. The paper seeks exact or nearly-exact solutions for long-lived asset prices that clarify the separate roles of risk aversion, the EIS, and the preference for early resolution of uncertainty.

Key idea

Decompose the price of a long-lived asset (equity) into an integral of equity strips (zero-coupon claims to dividends at each future date). Given the affine structure of the economy, each strip price is exponential-affine in the state variables, characterized by Riccati ODEs. The key insight: use log-linearization only once (for the wealth-consumption ratio), then price all other assets exactly conditional on that approximation. This yields solutions that are exact in two special cases: (1) no preference for early resolution of uncertainty (time-additive utility, theta=1), and (2) EIS = 1 (psi=1). The method is notably more accurate than standard double-linearization.

Method

  1. General model framework: continuous-time Markov endowment with Duffie-Epstein recursive utility, Brownian shocks, and Poisson jumps (rare disasters). State variables X_t follow an affine process.
  2. Value function: derive I(x) satisfying a nonlinear PDE; approximate log I(x) as affine in x (exact for psi=1 and theta=1).
  3. State-price density: express in terms of I(x), yielding risk prices for Brownian and Poisson shocks.
  4. Equity strip pricing: each strip H(D,x,tau) = D * exp(a_phi(tau) + b_phi(tau)‘x) where (a_phi, b_phi) solve Riccati ODEs that depend on the state-price density parameters.
  5. Total equity price: G(x) = integral_0^infty exp(a_phi(tau) + b_phi(tau)‘x) dtau. This is an integral of well-behaved exponential-affine functions, computed numerically.
  6. Risk premia: decompose into Brownian (covariance-based) and Poisson (rare-event) components; show that normal-times covariances are insufficient when rare events are present.

Results

  • Rare-event Consumption CAPM: risk premia include standard covariance terms plus terms for co-movement with marginal utility during disasters. The linear factor model breaks down with rare events.
  • Rare-event ICAPM: in the wealth-based model, risk prices for state variables depend on gamma > 1 (not gamma > 1/psi as in the consumption-based model).
  • Comparative statics cleanly separated: wealth-consumption ratio dynamics governed by EIS; cross-sectional risk premia (relative to consumption) governed by preference for early resolution; cross-sectional risk premia (relative to wealth) governed by risk aversion.
  • Solution accuracy: in an extension of the Wachter (2013) rare disaster model to non-unitary EIS, the strip-integral method is notably closer to the numerical solution than the standard double log-linearization.
  • Value premium: the paper shows that rare-event risk (covariance during extreme events, not normal-times covariance) can potentially explain the value premium.

Limitations

  • The wealth-consumption ratio still requires a log-linear approximation when psi != 1 and theta != 1; the method is not fully exact in the general case.
  • The continuous-time affine framework excludes some important model classes (e.g., stochastic volatility of volatility with non-affine structure).
  • The integral for total equity price must be computed numerically; no closed-form for the price-dividend ratio in general.
  • The paper is primarily theoretical; limited quantitative evaluation against data.

Open questions

  • How does the strip-integral approach extend to discrete-time Markov-switching models where the state space is finite?
  • Can the Riccati ODEs for strip prices be connected to the coupled Riccati recursions in regime-switching bond/asset pricing?
  • What is the quantitative impact of the single remaining approximation (wealth-consumption ratio) on estimated structural parameters?

My take

This paper provides the theoretical foundation for pricing long-lived assets as integrals of strips, each governed by Riccati dynamics. The connection to the CRE project is direct: the exponential-quadratic pricing factors in exponential-quadratic-asset-pricing-factors and the coupled Riccati recursions in coupled-riccati-recursions-regime-switching-asset are the discrete-time, regime-switching analogues of Tsai-Wachter’s continuous-time strip Riccati ODEs. The paper’s emphasis on the superiority of single-linearization over double-linearization resonates with the project’s Bansal approximation FAIL finding. The rare-event ICAPM results (Corollary 8) formalize why regime-switching risk premia cannot be reduced to normal-times covariances — exactly the mechanism at work in the CRE model where monetary policy regime shifts create non-Gaussian risk. The semigroup-approach-asset-pricing and principal-eigenfunction-pricing-operator concepts are closely related to the strip decomposition here.

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