Long-Term Risk: An Operator Approach

Problem

How do you decompose the long-horizon risk-return trade-off in a nonlinear continuous-time Markov environment, where the standard linearized log-normal apparatus (Hansen-Heaton-Li 2008) breaks down? Asset payoffs depend on both the future state and the date the payoff is realized, and risk-averse investors require horizon-dependent risk premia. Existing local (instantaneous) pricing fixes one end of the term structure of risk prices; the long end was poorly understood for nonlinear state dynamics. The paper builds an analytical apparatus that characterizes the limiting behavior of valuation operators as the payoff horizon goes to infinity, and connects that limit to a Perron-Frobenius eigenvalue/eigenfunction problem.

Key idea

Treat asset valuation as a one-parameter family (semigroup) of pricing operators

${\mathbb{M}}_t\psi (x)=E[M_t\psi (X_t)\mid X_0=x]$

indexed by horizon $t$, where $M$ is a strictly positive multiplicative functional of an underlying Markov process $X$. Find a positive eigenfunction $\varphi$ and real eigenvalue $\rho$ satisfying ${\mathbb{M}}_t\varphi =e^{\rho t}\varphi$. Then every multiplicative functional admits the canonical factorization

$M_t=e^{\rho t}{\hat{M}}_t\frac{\varphi (X_0)}{\varphi (X_t)}$

where $\hat{M}$ is a martingale, $\rho$ encodes the long-run risk-adjusted growth (or decay) rate, $\varphi$ encodes the limiting state-dependence, and the martingale $\hat{M}$ defines a distorted probability measure under which $X$ is stationary. Among potentially many positive eigenfunctions, exactly one — the one whose induced distortion is stochastically stable — characterizes long-run dominance, and its eigenvalue is the smallest (Proposition 7.2).

Method

1. Setup: Continuous-time strong Markov process $X$ on $D_0\subset {\mathbb{R}}^n$, semimartingale (Brownian + finite-rate jump). Pricing operators ${\mathbb{M}}_t$ form a semigroup whenever $M$ is multiplicative ($M_0=1$, $M_{t+u}=M_u({\theta }_t)M_t$).
2. Multiplicative parameterization: Strictly positive multiplicative functionals are exponentials of additive functionals $A_t={\int }_0^t\beta (X_u)du+{\int }_0^t\gamma (X_{u-}{)}^{\prime }d{B}_u+{\sum }_{0\le u\le t}\kappa (X_u,X_{u-})$, parameterized by $(\beta ,\gamma ,\kappa )$.
3. Extended generator: Differentiate the semigroup at $t=0$ to obtain an operator $\mathbb{A}$ that acts on functions of the state. The principal eigenvalue equation $\mathbb{A}\varphi =\rho \varphi$ becomes a tractable PDE/algebraic problem (Section 6.1: affine example).
4. Multiplicative factorization (Corollary 6.1): Once a positive eigenpair $(\rho ,\varphi )$ is found, define ${\hat{M}}_t=e^{-\rho t}{M}_t\varphi (X_t)/\varphi (X_0)$. Hansen and Scheinkman prove $\hat{M}$ is a (local) martingale; Appendix C gives sufficient conditions (Assumptions C.1–C.3) under which it is a true martingale.
5. Long-run dominance (Proposition 7.1): Under irreducibility (Assumption 7.3) and Harris recurrence (Assumption 7.4) of the distorted process, ${\lim }_{t\to \infty }{e}^{-\rho t}{\mathbb{M}}_t\psi =\varphi \int (\psi /\varphi )d\hat{\varsigma }$, where $\hat{\varsigma }$ is the stationary measure of the distorted Markov chain.
6. Uniqueness (Proposition 7.2): When multiple positive eigenfunctions exist, the relevant one is uniquely characterized as the eigenfunction whose distortion is stochastically stable; its eigenvalue is the smallest among admissible principal eigenvalues.
7. $L^p$ approximation (Propositions 7.3–7.4): Under a contraction (rho-mixing) assumption (7.5), the semigroup converges to its long-run limit at exponential rate $\eta >0$ in the appropriately weighted $L^p$ norm.
8. Existence (Section 9): Using a drift condition (Assumption 9.1: $\mathbb{A}V/V\le \underline{\mathbf{a}}$) plus an irreducibility condition on a resolvent operator $\mathbb{F}$ and a compactness/boundedness condition, the principal eigenfunction is constructed via Nummelin (1984)-style splitting and Kontoyiannis-Meyn (2003) techniques.
9. Applications (Section 8): long-run risk prices, term structure of bond yields, decomposition of returns to growth and value cash flows, holding-period limits.

Results

• Multiplicative factorization theorem: Every strictly positive multiplicative functional under the maintained assumptions decomposes as $M_t=e^{\rho t}{\hat{M}}_t\varphi (X_0)/\varphi (X_t)$. This separates the deterministic exponential trend, the permanent (martingale) risk component, and the transient (eigenfunction-ratio) state-dependence.
• $\rho$ is a long-run yield: For a stochastic discount factor with multiplicative functional $M$, $-\rho$ is the asymptotic continuously-compounded yield on a default-free zero-coupon bond, i.e., the long-run interest rate.
• Long-run risk prices: Sensitivities $d\rho /d\gamma$ to cash-flow risk exposures define the long-run risk-price vector. In the affine Feller-square-root + Ornstein-Uhlenbeck example (Example 6.2), the closed-form $\rho$ depends nonlinearly on the volatility-loading ${\gamma }_f$ via the quadratic discriminant, so long-run prices of volatility risk are nonlinear in exposure even when local prices are linear.
• Holding-period limit (Section 8): ${\lim }_{t\to \infty }E[S_t{D}_t/S_1\mid {\mathcal{F}}_1]/E[S_t{D}_t\mid {\mathcal{F}}_0]=e^{-\rho }{G}_1\varphi (X_1)/\varphi (X_0)$, decomposing the limiting return into eigenvalue, growth, and eigenfunction components — a clean continuous-time analog of Hansen-Heaton-Li (2008).
• Existence/uniqueness: Sufficient conditions (drift Lyapunov + irreducibility of the resolvent + boundedness of the constructed candidate $\varphi =\mathbb{G}s$) guarantee the principal eigenpair exists; among admissible eigenpairs, the stochastically stable one is unique up to scale (Proposition 7.2).
• Distorted-measure interpretation: The martingale $\hat{M}$ defines a probability change under which the original Markov process is stationary. The principal eigenvalue/eigenfunction simultaneously give the long-run growth rate, the long-run state dependence, and the right probability measure for long-horizon evaluation.

Limitations

• Computability: The paper proves existence and uniqueness but provides no general algorithm for $(\rho ,\varphi )$. Quasi-analytic solutions exist only for affine and a handful of other examples; for generic nonlinear environments, computing the principal eigenfunction is left open (explicitly flagged in the conclusions as the first major open problem).
• Finite-jump-rate restriction: The analysis assumes the underlying Markov process has finitely many jumps in any finite interval, which excludes most Lévy processes (infinite-activity jumps). The conclusions name extension to general Lévy processes as an open question.
• No refined approximation rate: Beyond the exponential bound $c{e}^{-\eta t}$ (Proposition 7.4), the paper does not characterize how fast $\eta$ depends on the model primitives, so it cannot tell you when the long-run approximation is good enough at a moderate horizon.
• Single-eigenvalue characterization: Only the dominant (Perron-Frobenius) eigenvalue is exploited. Subdominant eigenvalues — which control the rate of approach to the long-run limit — are not used, even though they would refine the approximation.
• Assumes a Markov state summary: The whole apparatus depends on a finite-dimensional Markov state summarizing all valuation-relevant information. Genuinely non-Markov environments (e.g., long-memory processes) fall outside the scope.

Open questions

• How to numerically compute the principal eigenpair $(\rho ,\varphi )$ for a general nonlinear continuous-time Markov environment?
• Can the apparatus be extended to multiplicative functionals built from infinite-activity Lévy processes?
• What are sharp bounds on the convergence rate $\eta$ in Proposition 7.4 in terms of model primitives?
• How can subdominant eigenvalues be used to obtain refined long-run approximations and to characterize the speed at which the long-run dominance kicks in?
• Does the multiplicative factorization survive under model misspecification — e.g., when the assumed Markov state is only an approximation?

My take

This paper is a load-bearing theoretical foundation for the CRE asset pricing project. The Riccati-based recursion in compute_quadratic_pricing_factors_msvar (and the geometric tail ${\eta }_{H}r/(1-r)$ used in the MC pricer’s $h$-burn extrapolation) is in spirit a numerical realization of the same exponential-trend-plus-transient decomposition. The Hansen-Scheinkman principal eigenvalue is exactly the long-run continuously-compounded yield that the recursion converges to as $T_{\stackrel{ˉ}{}}\to \infty$ — the convergence diagnostic “Riccati convergence: $A\to -\infty$ is correct” in the project notes is the eigenvalue equation written in coefficient form. Any operator-floor null hypothesis (e.g., the leading post-Exp 12 candidate N3, that the geometric tail with a single global $r$ is the real bottleneck) is implicitly an assertion about the dominant eigenvalue and its rate-of-approach $\eta$. The fact that this paper proves existence/uniqueness only and explicitly flags the absence of a general numerical algorithm explains why the project’s computation of long-run yields in regime-switching environments is so much harder than in the closed-form affine examples in Section 6.1: the model is past Hansen-Scheinkman’s quasi-analytic frontier.

Related

• Concept: principal-eigenfunction-pricing-operator
• Concept: multiplicative-factorization-stochastic-discount-factor
• Concept: semigroup-approach-asset-pricing
• Topic: commercial-real-estate-pricing-regimes
• Topic: markov-switching-term-structure-models
• Author: lars-peter-hansen
• Author: jose-scheinkman
• supports: sdf-multiplicative-factorization-decomposes-long-run
• supports: principal-eigenvalue-determines-long-run-asset