Multiplicative Factorization of the Stochastic Discount Factor

Definition

A strictly positive multiplicative functional $M$ — typically a stochastic discount factor process $S_t$, a cash-flow growth process $G_t$, or their product — built on a Markov state $X$ admits the canonical factorization

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

where $\rho$ is a real scalar (the principal eigenvalue), $\hat{M}$ is a martingale whose logarithm has stationary increments, and $\varphi$ is a strictly positive function on the state space (the principal eigenfunction). The three components are interpreted as a deterministic exponential trend ($e^{\rho t}$), a permanent risk component ($\hat{M}$), and a transient state-dependent ratio ($\varphi (X_0)/\varphi (X_t)$).

Intuition

The decomposition separates “what grows forever” from “what eventually averages out.” The trend $e^{\rho t}$ captures the long-run yield of the discount factor; the martingale $\hat{M}$ captures permanent shocks that never wash out (and defines the right change of measure under which the state process is stationary); the transient factor $\varphi (X_0)/\varphi (X_t)$ captures everything that depends on the current Markov state but vanishes in long-horizon expectations. For a stochastic discount factor, $-\rho$ is the asymptotic continuously-compounded risk-free yield and $\hat{M}$ encodes the “long-run risk” that earns the long-run risk premium.

Formal notation

If $M$ is a strictly positive multiplicative functional and $(\rho ,\varphi )$ is a principal eigenpair of the associated semigroup, define

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

Hansen and Scheinkman (2009, Corollary 6.1) prove $\hat{M}$ is a local martingale; Appendix C gives sufficient conditions (Assumptions C.1–C.3) under which it is a true martingale. The $\hat{M}$-induced probability measure makes $X$ stationary (under Assumption 7.2), allowing standard ergodic-Markov tools to characterize long-horizon behavior.

Variants

• Alvarez-Jermann (2005) version: A discrete-time precursor that proposed the same decomposition but without the formal link to principal eigenvalues; Hansen-Scheinkman (footnote 3) note Alvarez-Jermann cited an early version of the present paper for that link.
• Affine-state closed form: For exponential-affine SDFs on Feller-square-root + OU states, $\varphi (x)=\exp (c_f{x}_f+c_o{x}_o),{\hat{M}}_t$ has an explicit Itô representation, and the distorted drift is ${\xi }_f{\bar{x}}_f\pm x_f\sqrt{({\xi }_f-{\gamma }_f{\sigma }_f{)}^2-{\sigma }_f^2(2{\beta }_f+{\gamma }_f^2)}$; only the negative-root branch is stochastically stable (Hansen-Scheinkman Section 7.2.1).
• Cash-flow growth version: Apply the decomposition to $M=GS$ where $G$ is a multiplicative growth functional, giving the limiting holding-period return $e^{-\rho }{G}_1\varphi (X_1)/\varphi (X_0)$ (Hansen-Scheinkman Section 8).

Comparison

• vs Beveridge-Nelson decomposition: Both separate permanent and transitory components, but Beveridge-Nelson is additive and linear, while Hansen-Scheinkman is multiplicative and nonlinear, designed for strictly positive processes such as discount factors and cash-flow growth.
• vs Local risk-neutral measure: The local risk-neutral measure makes discounted prices martingales over arbitrary horizons; the Hansen-Scheinkman martingale $\hat{M}$ instead defines a measure under which the state process is stationary, which is the right object for long-horizon limits.

When to use

• Identifying the long-run yield $-\rho$ of any strictly positive multiplicative functional in a Markov environment.
• Constructing the change of measure under which a nonstationary discount factor becomes a (martingale) tractable for long-horizon asset pricing.
• Defining permanent vs transitory components of a stochastic discount factor for the long-run risk literature (Bansal-Yaron, Hansen-Heaton-Li).
• Decomposing limiting holding-period returns in models with stochastic cash-flow growth.

Known limitations

• The decomposition is not unique without further restrictions: multiple positive eigenfunctions can each generate a candidate $\hat{M}$. Hansen-Scheinkman Proposition 7.2 selects the stochastically stable one as canonical.
• Verifying that the candidate $\hat{M}$ is a true martingale (not just local) requires checking technical conditions (Hansen-Scheinkman Appendix C) that are nontrivial outside specific parametric families.
• Computing $(\rho ,\varphi )$ in non-affine environments has no general algorithm.

Open problems

• General-purpose numerics for the multiplicative factorization in regime-switching, jump, or genuinely nonlinear environments.
• Robustness of the decomposition under model misspecification.
• Connection to subdominant eigenvalues for refined finite-horizon corrections.

Key papers

• hansen-scheinkman-2009-long-term-risk-operator-approach — proves existence, uniqueness, and the link to principal eigenvalues; gives the semigroup machinery and the long-run dominance result.

My understanding

In the CRE asset pricing project, the decomposition

$\text{(quadratic-Riccati price)}=\text{(geometric tail)}+\text{(transient correction)}$

used in mc_pricer.jl and mc_tail.jl is structurally a Hansen-Scheinkman factorization with the transient correction playing the role of $\varphi (X_0)/\varphi (X_t)$ and the geometric tail ${\eta }_{H}r/(1-r)$ playing the role of $e^{\rho t}$. The MC pricing pipeline’s per-eval rebuild (rather than fixed lookup) is in effect re-estimating the eigenvalue $\rho$ at each $\theta$, which is why a fixed lookup drifts catastrophically (+5269 nats): a stale $\rho$ misprices the dominant exponential-trend component. The Apartment “slow asset” gotcha (needing $T_{\stackrel{ˉ}{}}\approx 50K$ at the new MAP) is the same statement at the level of finite-horizon convergence: the $L^p$ approximation rate $\eta$ in Hansen-Scheinkman Proposition 7.4 is small for that asset, so the transient component decays slowly.