The principal eigenvalue of the pricing semigroup determines the long-run yield and the long-run risk-price vector

Statement

For a strictly positive multiplicative functional $M$ of a continuous-time strong Markov process whose principal eigenpair $(\rho ,\varphi )$ exists and is stochastically stable, the eigenvalue $\rho$ is the long-run continuously-compounded growth (or decay) rate of the family of pricing operators ${\mathbb{M}}_t$. Specifically:

1. Long-run yield: ${\lim }_{t\to \infty }{t}^{-1}\log {\mathbb{M}}_t\psi (x)=\rho$ for any payoff $\psi$ in an appropriately weighted space.
2. Long-run risk price vector: For families of multiplicative functionals indexed by cash-flow risk exposures $\gamma$, the long-run risk price of a unit exposure to a risk source is $d\rho /d\gamma$ — the sensitivity of the principal eigenvalue with respect to the exposure parameter.
3. Holding-period limit: ${\lim }_{t\to \infty }E[S_t{D}_t/S_1\mid {\mathcal{F}}_1]/E[S_t{D}_t\mid {\mathcal{F}}_0]=e^{-\rho }{G}_1\varphi (X_1)/\varphi (X_0)$, decomposing the limiting return into eigenvalue, growth, and eigenfunction-ratio components.

Evidence summary

Hansen-Scheinkman (2009) Proposition 7.1 establishes the limit ${\lim }_{t\to \infty }{e}^{-\rho t}{\mathbb{M}}_t\psi =\varphi \int (\psi /\varphi )d\hat{\varsigma }$, which is the formal sense in which $\rho$ is the long-run growth rate. Section 8 then derives the long-run risk-price formulas in the affine Feller-square-root + Ornstein-Uhlenbeck example: $d\rho /d{\gamma }_o^s=-{\gamma }_o^r-({\beta }_o^s/{\xi }_o){\sigma }_o$ for the Brownian-exposure $B_o$, and a nonlinear (because $c_f$ depends on ${\gamma }_f$ through a quadratic) formula for the $B_f$ exposure. The persistence parameter ${\xi }_o$ of the Ornstein-Uhlenbeck factor governs how strongly cash-flow risk feeds into the long-run price, recovering the discrete-time log-normal analog of Hansen-Heaton-Li (2008). Proposition 7.4 strengthens the limit to exponential $L^p$ convergence at rate $\eta >0$, providing the quantitative justification for using $\rho$ as a long-run yield at moderate horizons.

Conditions and scope

• Hansen-Scheinkman Assumptions 6.1, 7.1–7.4 (existence of a true-martingale candidate distortion; irreducibility and Harris recurrence of the distorted process under a stationary measure $\hat{\varsigma }$).
• A stochastic stability selection rule must be applied to choose the principal eigenpair when multiple positive eigenfunctions exist (Proposition 7.2). The selected eigenvalue is the smallest admissible one.
• Sensitivity formulas for long-run risk prices ($d\rho /d\gamma$) require the eigenvalue to be smoothly differentiable in the exposure parameter — generally true in affine examples and more broadly under standard regularity, but not given a general perturbation theorem in Hansen-Scheinkman 2009.
• The long-run risk-price vector from the eigenvalue sensitivity $d\rho /d\gamma$ generally differs from the local risk-price vector implied by the instantaneous valuation operator: they coincide for log-normal specifications but not in general (Hansen-Scheinkman Section 8).

Counter-evidence

None known within the assumed framework. Open questions center on: (a) the convergence rate $\eta$ (how fast $\rho$ describes finite-horizon behavior); (b) the gap between long-run risk prices and finite-horizon risk prices (the “term structure of risk prices”); (c) extensions to environments where the principal eigenpair does not exist or is not unique.

Linked ideas

(none yet — open for follow-up)

Open questions

• Sharp model-primitive characterization of the convergence rate $\eta$ in Hansen-Scheinkman Proposition 7.4: when is the long-run approximation accurate at moderate horizons (years rather than centuries)?
• For regime-switching models, how does the principal eigenvalue and its long-run risk-price gradient behave as a function of the regime-switching rates?
• How do subdominant eigenvalues correct the long-run yield approximation at finite horizons?