6. Multiplicative Functionals

Authors: Jaroslav Borovicka, Lars Peter Hansen, and Thomas Sargent

Date: November 2024

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Chapter Processes with Markovian increments described additive functionals of a Markov process. This chapter describes exponentials of additive functionals that we call multiplicative functionals and that we can use to model stochastic growth, stochastic discounting, and their interactions. After adjusting for geometric growth or decay, a multiplicative functional contains a martingale component that turns out to be a likelihood ratio process that is itself a special type of multiplicative functional called an exponential martingale. We shall see how a likelihood ratio process is used to construct an alternative probability model with Markov dynamics by simply multiplying an original probability measure by the likelihood ratio process. This procedure is widely used in asset pricing theory and helps us to represent stochastic components of growth and discounting that persist over long horizons. It is also plays an essential role in statistical discrimination. We will have several other applications of multiplicative functionals, including returns and positive cash flows that compound over multiple horizons, cumulative stochastic discount factors used represent prices of such multi-period cash flows, and subjective beliefs of private sector investors and policy makers that might deviate from rational expectations. To analyze multiplicative functionals, we apply mathematical tools closely related to tools from the statistical theory of large deviations.

6.1. Geometric growth and decay

To construct a multiplicative functional, we start with an underlying Markov process \(X\) that has stationary distribution \(Q\).

Definition 6.1

Let \(Y \eqdef \{ Y_t\}\) be an additive functional that as in Chapter Processes with Markovian increments is described by \(Y_{t+1} - Y_t = \kappa(X_t, W_{t+1})\), where \(X_t\) is the time \(t\) component of a Markov state vector and \(W_{t+1}\) is the time \(t+1\) value of a martingale difference process (\({\mathbb E} \left(W_{t+1} \mid {\mathcal A}_t \right) = 0 \)) of unanticipated shocks. We say that \(M \eqdef \{M_t: t \ge 0 \} = \{ \exp(Y_t) : t \ge 0 \}\) is a multiplicative functional parameterized by \(\kappa\). When \(Y_0\) is a (Borel measurable) function of \(X_0\), \(M_0 >0\) is also a (Borel measurable) function of \(X_0\).

An additive functional grows or decays linearly, so the exponential of an additive functional grows or decays geometrically. In Chapter Processes with Markovian increments, we constructed a Law of Large Numbers and a Central Limit Theorem for additive functionals. In this chapter, we use other mathematical tools to analyze the limiting behavior of multiplicative functionals. In what follows, we refer to

\[N_{t+1} = \frac {M_{t+1}}{M_t}\]

as the multiplicative increment of the multiplicative process \(M\).

6.2. Special multiplicative functionals

We define the three primitive types of multiplicative functionals.

Example 6.1

Suppose that \(\kappa = \eta\) is constant and that \(M_0\) is a Borel measurable function of \(X_0\). Then

\[M_t = \exp\left( t \eta \right)M_0.\]

This process grows or decays geometrically.

Example 6.2

Suppose that

\[{\mathbb E} \left[ \exp \left[ \kappa \left( X_t, W_{t+1} \right) \right] \vert X_t \right] = 1.\]

Then

(6.1)\[{\mathbb E} \left(M_{t+1} \vert \mathcal{A}_t \right) = M_t \]

and hence,

\[{\mathbb E} \left(N_{t+1} \vert \mathcal{A}_t \right) = 1\]

A multiplicative functional that satisfies (6.1) is called a multiplicative martingale. In what follows, we denote such processes as \(M = L\) because, as we will see, we may view it as a likelihood ratio processes.

Example 6.3

Suppose that \(M_t = \exp\left[h(X_t)\right]\) where \(h\) is a Borel measurable function. The associated additive functional satisfies

\[\begin{align*} Y_{t+1} - Y_t & = \log M_{t+1} - \log M_t \cr & = h(X_{t+1}) - h(X_t) \cr & = h\left[ \phi(X_t, W_{t+1} ) \right] - h(X_t) \end{align*}\]

and so is parameterized by \(\kappa(X_t, W_{t+1}) = h\left[ \phi(X_t, W_{t+1} ) \right] - h(X_t)\) with initial condition \(Y_0 = h(X_0)\).

When the process \(\{X_t\}\) is stationary and ergodic, multiplicative functional Example 6.1 displays expected growth or decay, while multiplicative functionals Example 6.2 and Example 6.3 do not. Multiplicative functional Example 6.3 is stationary, while Example 6.1 and Example 6.2 are not.

We can construct other multiplicative functionals simply by multiplying two or more instances of these primitive ones. In the following section, we reverse that process: we take an arbitrary multiplicative functional and (multiplicatively) decompose it into instances of our three types of multiplicative functionals. Before doing so, we explore multiplicative martingales in more depth.

6.3. Multiplicative martingales and likelihood processes

Multiplicative martingales induce alternative probabilities and are thus likelihood-ratio processes. Specifically, we use a multiplicative martingale to represent an alternative probability model. We can characterize such a model with a set of implied conditional expectations of all bounded random variables, \(B_{t+1},\) that are measurable with respect to \({\mathfrak A}_{t+1}\). The constructed conditional expectation is

(6.2)\[E \left(N_{{t+1}} B_{{t+1}} \mid {\mathfrak A}_t \right) . \]

The multiplication of \(B_{t+1}\) by \(N_{t+1}\) changes the baseline probability to an alternative model. To serve this purpose the random variable \(N_{t+1}\) must satisfy:

1. \(N_{t+1} \ge 0\);

2. \(E\left(N_{t+1} \mid {\mathfrak A}_t \right) = 1\);

3. \(N_{t+1}\) is \({\mathfrak A}_{t+1}\) measurable.

Property 1 is satisfied because conditional expectations map positive random variables \(B_{t+1}\) into positive random variables that are \({\mathfrak A}_t\) measurable. Properties 2 and 3 are satisfied because of the \(N\) is the multiplicative increment of a multiplicative martingale. The resulting process \(L\) can be viewed as a likelihood ratio process of for the alternative process expressed in terms of the baseline process.

This way of representing an alternative probability model is restrictive. Thus, if a nonnegative random variable has conditional expectation zero under the baseline probability, it will also have zero conditional expectation under the alternative probability measure, a version of absolute continuity here applied to transition probabilities. From a statistical perspective, violating absolute continuity would make possible model decision rules that correctly select models with full confidence from only finite samples.

Multiplicative martingales provide a way to model the subjective beliefs of private agents or policy-makers within dynamic, stochastic equilibrium models when these beliefs are allowed to depart from rational expectations.

The following are examples of multiplicative martingales constructed from probability models specified in more familiar ways.

Example 6.4

Consider a baseline Markov process having transition probability density \(\pi_o\) with respect to a measure \(\lambda\) over the state space \(\mathcal{X}\)

\[P_o(dx^+|x) \lambda(dx^+) = \pi_o(x^+ \mid x) \lambda(dx^+)\]

Let \(\pi\) denote some other transition density that we represent as

\[\pi(x^+ \mid x) \lambda(dx^+ ) = \left[ {\frac {\pi(x^+ \mid x)}{\pi_o(x^+ \mid x)}}\right] \pi_o(x^+ \mid x) \lambda(dx^+ )\]

where we assume that \(\pi_o(x^+ \mid x) = 0\) implies that \(\pi(x^+ \mid x) = 0\) for all \(x^+\) and \(x\) in \(\mathcal{X}\).
Construct the multiplicative increment process as:

\[N_{t+1} = {\frac {\pi(X_{t+1} \mid X_t)}{\pi_o(X_{t+1} \mid X_t)}} .\]

Example 6.5

Suppose that \(X\) evoles as a vector-autoregression:

\[X_{t+1} = {\mathbb A} X_t + {\mathbb B} W_{t+1} \]

where \({\mathbb A}\) is a stable matrix, \(\{W_{t+1} : t \ge 0 \}\) is an i.i.d. sequence of \({\cal N}(0,I)\) random vectors conditioned on \(X_0\). and \({\mathbb B}\) is a square, nonsingular matrix. Assume a baseline model that has the same functional form with particular settings of the parameters that appear in the matrices \(({\mathbb A}_o, {\mathbb B}_o)\) Let \(N_{t+1}\) be the one-period conditional log-likelihood ratio

\[\begin{split}\log N_{t+1} = - {\frac 1 2} (X_{t+1} - {\mathbb A} X_t)'\left({\mathbb B}{\mathbb B}'\right)^{-1}(X_{t+1} - {\mathbb A} X_t) \\ + {\frac 1 2} \left(X_{t+1} - {\mathbb A}_o X_t \right)'\left({\mathbb B}_o{{\mathbb B}_o}'\right)^{-1} \left(X_{t+1} - {\mathbb A}_o X_t \right) \\ - {\frac 1 2} \log \det \left({\mathbb B}{\mathbb B}'\right) + {\frac 1 2} \log \det \left( {\mathbb B}_o{{\mathbb B}_o}'\right)\end{split}\]

Notice how we have subtracted components coming from the baseline model.

Remark 6.1

In the model in Example 6.5 there are the same number of shocks, i.e., entries of \(W\), as there are components of the observation vector, i.e., entries of \(X\). Because \(\mathbb{B}\) is a nonsingular square matrix. A more general starting point a hidden Markov state model like that presented in Section Kfilter with a time-invariant innovations representation conditions on an infinite past of observations. Actual statistical analyses often condition only on a finite past when forming likelihood functions. That typically produces an \(N_{t+1}\) process that shares asymptotic properties with an alternative process that comes from conditioning on an infinite past.

As shown by [Hansen and Scheinkman, 2009], multiplicative martingales also offer a way to study the valuation of cumulative returns. Let \(R_t\) be a multiplicative process that measures the cumulative return between date \(t\) and date zero. Let \(S_t\) be corresponding equilibrium discount factor between these same two dates. Then equilibrium pricing implies that \(L = RS\) is a multiplicative martingale. This follows from the familiar one-period relations:

\[{\mathbb E} \left[ \left( \frac {S_{t+1}}{S_t} \right)\left( \frac {R_{t+1}}{R_t}\right) \mid {\mathfrak A}_t \right] = 1 \]

where \(S_{t+1}/{S_t}\) is the one-period stochastic discount factor and \(R_{t+1}/{R_t}\) is the one-period gross return.

To reveal a limiting behavior of multiplicative martingales, apply Jensen’s inequality for the concave function, \(\log L,\) illustrated in Fig. 6.1.

Fig. 6.1 Jensen’s Inequality. The logarithmic function is the concave function that equals zero when evaluated at unity. An interior average of the endpoints of the straight line lies below the logarithmic function. Jensen’s Inequality asserts that the line segment lies below the logarithmic function.

By Jensen’s inequality,

\[{\mathbb E} \left(\log N_{t+1} \mid {\mathfrak A}_t \right) \le \log {\mathbb E} \left(N_{t+1} \mid {\mathfrak A}_t \right) = 0.\]

Normalize \(L_0=1\) and form

\[L_t = \prod_{\tau=1}^t N_\tau .\]

Note that

\[\log L_t = \sum_{\tau = 1}^t \log N_{\tau} , \]

and thus

\[E \left(\log L_{t+1} \mid {\mathfrak A}_t \right) \le \log L_t.\]

This implies that under the baseline model the log-likelihood ratio process \(L\) is a super martingale relative to the information sequence \(\{ {\mathfrak A}_t : t\ge 0\}\).

From the Law of Large Numbers, the population mean is well approximated by a sample average from a long time series. That opens the door to discriminating between two models. Under the baseline model, the log likelihood ratio process scaled by \(1/t\) converges to a negative number. After changing roles of the baseline and alternative models, we can do an analogous calculation that entails using \({\frac 1 {N_{t+1}} }\) instead of \(N_{t+1}\) as an increment. Then the scaled-by-\(1/t\) log likelihood ratio would converge to the expectation of \(- \log N_{t+1}\) under the alternative model that is now in the denominator of the likelihood ratio. This limit would be positive under the assumption that the alternative model generated the data. These calculations justify selecting between the two models by calculating \(\log L_{t}\) and checking if it is positive or negative. This procedure amounts to a special case of the method of maximum likelihood.

Suppose that data are not generated by the baseline model. Instead, imagine that the statistical model implied by the change of measure \(N_{t+1}\) governs the stochastic evolution of the observations. Define conditional entropy of the martingale increment relative to baseline model as the following conditional expectation:

\[E \left( N_{t+1} \log N_{t+1} \mid {\mathfrak A}_t \right) \ge 0\]

To understand the right-side inequality, the multiplication of \(\log N_{t+1}\) by \(N_{t+1}\) changes the conditional probability distribution from the misspecified baseline model to the alternative statistical model that we assume generates the data. The function \( n \log n\) is convex and equal to zero for \(n=1\). Therefore, Jensen’s inequality implies that conditional relative entropy is nonnegative and equal to zero when \(N_{t+1} = 1\). Notice that

\[\begin{align} {\mathbb E} \left(L_{t+1} \log L_{t+1} \mid {\mathfrak A}_t \right) & = L_t {\mathbb E} \left( N_{t+1} \log N_{t+1} \mid {\mathfrak A}_t \right) + L_t \log L_t \cr & \ge L_t \log L_t. \end{align} \]

Thus \(L \log L\) is a sub martingale. The expression

\[{\mathbb E} \left( L_t \log L_t \mid {\mathfrak A}_0 \right) \ge 0,\]

and is a measure of relative entropy over a \(t\)-period horizon. Relative entropy is often used to analyze model misspecifications. It is also a key component for studying the statistical theory of “large deviations” for Markov processes, as we shall discuss later.

Finally, consider a special case of Bayes’ Law. A decision-maker does not know which model generates the data.
Attach a subjective prior probability \(\pi_o\) to the baseline probability model with probability \(1 - \pi_o\) on the alternative. Suppose that \(L\) is a likelihood ratio process with \(L_t\) reflecting information available at date \(t\). Then the date \(t\) posterior probability for the baseline and alternative probability models are:

\[{\frac {\pi_o}{L_{t} (1-\pi_o) + \pi_o}} \quad {\textrm{and}} \quad \frac {L_{t}(1 - \pi_o)}{L_{t} (1-\pi_o) + \pi_o} .\]

When \({\frac 1 {t}} \log L_{t}\) converges to a negative number , the first probability converges to one. In contrast, when \({\frac 1 {t}} \log L_{t}\) converges to a positive number under the alternative probability the second probability converges to one. When the data are generated by the baseline probability model, the Law of Large Numbers implies the former; and when the data are generated by the alternative probability model, the Law of Large Numbers implies the latter. There is a direct extension of this analysis under which some other model generates the data.

6.4. Factoring multiplicative functionals

Following [Hansen and Scheinkman, 2009] and [Hansen, 2012], we factor a multiplicative functional into three multiplicative components having the primitive types Example 6.1, Example 6.2, Example 6.3. As in definition Definition 6.1, let \(Y\) be an additive functional, and let \(M = \exp(Y)\). Apply a one-period operator \(\mathbb{M}\) defined by

(6.3)\[\begin{align} \mathbb{M}f(x) & \eqdef {\mathbb E}\left[ \exp(Y_{t+1} - Y_t) f(X_{t+1}) \mid X_t = x \right] \cr & = {\mathbb E}\left[ \left(\frac{M_{t+1}}{M_t} \right) f(X_{t+1}) \vert X_t = x \right]. \end{align}\]

to bounded Borel measurable functions \(f\) of the Markov state. By applying the Law of Iterated expectations, a two-period operator iterates \(\mathbb{M}\) twice to obtain:

(6.4)\[\begin{align} \mathbb{M}^2 f(x) & \eqdef {\mathbb E} \left[ \exp(Y_{t+2} - Y_t) f(X_{t+2}) \mid X_t = x \right] \cr & = {\mathbb E}\left[ \left(\frac{M_{t+2}}{M_t} \right) f(X_{t+2}) \mid X_t = x \right], \end{align} \]

with corresponding definitions of \(j\)-period operators \(\mathbb{M}^j\). The family of operators is a special case of what is called a ``semi-group.’’ The domain of the semigroup can typically be extended to a larger family of functions, but this extension depending on further properties of the multiplicative process used to construct it. For an extended application to asset valuation and investor preferences, see []. We will explore these applications in discussions that follow.

First, for a strictly positive \(f\), construct the limit

(6.5)\[\begin{align*} {\tilde \eta} &= \lim_{j\rightarrow \infty} \frac{1}{j} \log E\left[ \exp(Y_{t+j} - Y_t) f(X_{t+j}) \mid X_t = x \right] \cr &= \lim_{j\rightarrow \infty} \frac{1}{j} \log E\left[ \frac{M_{t+j}}{M_t} f(X_{t+j}) \mid X_t = x \right] \cr &= \lim_{j\rightarrow \infty} \frac{1}{j} \log \mathbb{M}^j f(x) \end{align*}\]

when the limit is finite. For instance, \(f\) could be identically one. We call \({\tilde \eta}\) the asymptotic growth (or decay) rate of the multiplicative functional \(M\). Multiplying the multiplicative functional by \(\exp(-\eta t)\) removes expected asymptotic growth from the semigroup.

To refine this limiting characterization of a multiplicative functional and obtain two other components of the factorization, we apply what is referred to mathematics as Perron-Frobenius theory. We start by posing:

Eigenvalue-eigenfunction Problem: Solve

(6.6)\[\mathbb{M}{e}(x) = \exp\left( {\tilde \eta} \right) {\tilde e}(x)\]

for an eigenvalue \(\exp(\tilde \eta)\) and a positive eigenfunction \({\tilde e}\).

We call the largest eigenvalue, the principal eigenvalue, and the associated eigenvector the principal eigenfunction of the operator \(\mathbb{M}\). A positive eigenfunction \({\tilde e}\) is a function of the Markov state that can be expected to grow (or decay) geometrically at the long-run growth rate \(\eta = {\tilde \eta}\). Write the eigenfunction equation (6.6) as:

\[E\left[ {\frac {{ M}_{t+1}}{{ M}_t}} {\tilde e}(X_{t+1}) \vert X_t \right] = \exp\left( {\tilde \eta} \right) {\tilde e}(X_t).\]

Iterating the eigenfunction equation implies

\[E\left[ {\frac {{ M}_{t+j}}{{ M}_t}} {\tilde e}(X_{t+j}) \vert X_t \right] = \exp\left( { j \tilde \eta} \right) {\tilde e}(X_t). \]

Solve for the principal eigenvalue and eigenvector, and define:

\[{\widetilde N}_{t+1} \eqdef \exp(- {\tilde \eta}) {\frac {M_{t+1}{\tilde e}(X_{t+1})}{M_t {\tilde e}(X_t)}} = \exp\left[{\tilde \kappa}(X_t, W_{t+1}) \right],\]

and build

\[{\widetilde L}_{t+1} = {\widetilde N}_{t+1} {\widetilde L}_t , \hspace{.3cm} {\widetilde L}_0 = 1. \]

By construction, \({\widetilde N}_{t+1}\) has a conditional expectation equal to unity. Consequently, \( {\widetilde L}\) is a multiplicative martingale.

Theorem 6.1

Let \(M_t\) be a multiplicative functional. Suppose that the principal eigenvalue-eigenfunction Problem has a solution with principal eigenfunction \({\tilde e}(X)\). Then the multiplicative functional is the product of three components that are instances of the primitive functionals in examples Example 6.1, Example 6.2, and Example 6.3:

(6.7)\[{\frac {M_t}{ M_0}} = \exp\left( {\tilde \eta} t \right) {\widetilde L}_t\left[ {\frac {{\tilde e}(X_0)}{{\tilde e}(X_t) }} \right]\]

where \({\widetilde L}_t\) is a multiplicative martingale.

The factorization of a multiplicative functional described in Theorem 6.1 is a counterpart to the Proposition 4.1 decomposition of an additive functional. We used the martingale in Proposition 4.1 to identify the permanent component of an additive functional in Chapter Processes with Markovian increments. In this chapter, we shall use the multiplicative martingale isolated by Theorem 6.1 to represent a change of probability measure. The fact that the additive martingale \(Y= \log(M)\) has a variance that grows linearly over time contributes a component to the exponential trend of the multiplicative functional \(M\) along with a martingale component. The following log-linear, log-normal model displays relevant mechanics.

Example 6.6

Consider a stationary \(X\) process and an additive \(Y\) process described by the VAR

\[\begin{align*} X_{t+1} & = {\mathbb A} X_t + {\mathbb B} W_{t+1} \cr Y_{t+1} - Y_t & = \nu + {\mathbb D} \cdot X_t + {\mathbb F} \cdot W_{t+1} \end{align*}\]

where \({\mathbb A}\) is a stable matrix and \(\{ W_{t+1} : t \ge 0 \}\) is a sequence of independent and identically normally distributed random vectors with mean zero and covariance matrix \({\mathbb I}\). In Proposition Proposition 4.1 of Chapter Processes with Markovian increments, we described the decomposition

(6.8)\[Y_{t} - Y_0 = t \nu + \left[\sum_{j=1}^{t} {\mathbb H} \cdot W_{j}\right] - g(X_{t-1 }) + g(X_0)\]

where

\[\begin{align*} {\mathbb H} = & {\mathbb F} + {\mathbb B}'\left({\mathbb I} - {\mathbb A}' \right)^{-1} {\mathbb D} \cr g(x) = & {\mathbb D}'\left({\mathbb I} - {\mathbb A}\right)^{-1} x. \end{align*}\]

Let \(M_t = \exp(Y_t)\). Use equation (6.8) to deduce

\[\frac{M_t}{M_0} = \exp\left( \tilde \eta t \right) {\widetilde L}_t \left[ \frac{\tilde e (X_0)} {\tilde e(X_t)} \right]\]

where

\[\tilde \eta = \nu + \frac{{\mathbb H} \cdot {\mathbb H} }{2} ,\]

(6.9)\[\widetilde N_{t+1} = \exp \left( {\mathbb H} \cdot W_{t+1} -\frac{ {\mathbb H} \cdot {\mathbb H} }{2} \right), \quad \widetilde L_0 =1 ,\]

and

\[\tilde e(x) = \exp[g(x)] = \exp \left[ {\mathbb D}' \left({\mathbb I} - {\mathbb A} \right)^{-1} x \right] \]

The martingale component to the multiplicative functional has “unusual behavior” as a stochastic process. It has expectation one by construction. By the Martingale Convergence Theorem it is guaranteed to converge, typically to zero as is evident in Fig. 6.2, which plots the probability density function for different values of \(t\).

Fig. 6.2 Density of \(\widetilde{L}_t\) for different values of \(t\).

As we have emphasized, we are more interested in this martingale component as a change of probability measure. Given formula for \({\widetilde N}_t,\) the change in probability measure induces mean \( {\mathbb H}\) in the conditional distribution for the shock \(W_{t+1}\).

The case in which \(X_t\) is a finite-state Markov chain is also manageable computationally. In this case the principal eigenvalue calculation collapses to finding an eigenvector of a matrix with all positive entries.

Example 6.7

The stochastic process \(X_t\) is governed by a finite state Markov chain on state space \( \{ {\sf s}_1, {\sf s}_2, \ldots, {\sf s}_n \}\), where \(s_i\) is the \(n \times 1\) vector whose components are all zero except for \(1\) in the \(i^{th}\) row. The transition matrix is \({\mathbb P},\) where \({\sf p}_{ij} = \textrm{Prob}( X_{t+1} = [{\sf s}_j | X_t = {\sf s}_i)\). We can represent the Markov chain as

\[X_{t+1} = {\mathbb P}' X_t + W_{t+1}\]

where \({\mathbb E} (X_{t+1} | X_t ) = {\mathbb P}' X_t \), \({\mathbb P}'\) denotes the transpose of \(P\), and \(\{W_{t+1}\}\) is an \(n \times 1\) vector process that satisfies \({\mathbb E} ( W_{t+1} | X_t) = 0 \), which is therefore a martingale difference sequence adapted to \(X_t, X_{t-1}, \ldots , X_0\).

Let \({\mathbb G}\) be an \(n \times n\) matrix whose \((i,j)\) entry \({\sf g}_{ij}\) is an additive net growth rate \(Y_{t+1} - Y_t\) experienced when \(X_{t+1} = {\sf s}_j\) and \(X_t = {\sf s}_i\). The stochastic process \(Y\) is governed by the additive functional

\[Y_{t+1} - Y_t = (X_t)'{\mathbb G} X_{t+1} .\]

Let \(M= \exp(Y)\). Define a matrix \({\mathbb M}\) whose \((i,j)^{th}\) element is \({\sf m}_{ij} = \exp({\sf g}_{ij}).\) The stochastic process \(M\) is governed by the multiplicative functional:

(6.10)\[{\frac {M_{t+1}}{M_t}} = \exp\left[ (X_t)' {\mathbb G} X_{t+1} \right] = (X_t)'{\mathbb M} X_{t+1}.\]

Associated with this multiplicative functional is the principal eigenvalue problem

\[{\mathbb E} \left[ {\frac {M_{t+1}}{M_t}} \tilde e \cdot X_{t+1} | X_t = x \right] = \exp\left( \tilde \eta \right) \tilde e \cdot x.\]

To convert this to a matrix problem, write the \(j^{th}\) entry of \({\tilde e}\) as \({\tilde e}_j\). Since \(X_t\) always assumes the value of one of the coordinate vectors \({\sf s}_i, i =1, \ldots, n\),

\[(X_t)'{\sf M} X_{t+1} = {\sf m}_{ij}\]

when \(X_t = {\sf s}_i\) and \(X_{t+1} = {\sf s}_j\). This allows us to rewrite the principal eigenvalue problem as

\[\sum_j {\sf p}_{ij} {\sf m}_{ij} \tilde e_j = \exp(\tilde \eta) \tilde {\sf e}_i\]

or

(6.11)\[\widetilde {\mathbb P} \tilde e = \exp(\tilde \eta) \tilde e\]

where \(\widetilde {\sf p}_{ij} = {\sf p}_{ij} {\sf m}_{ij}\) and \({\sf e}_i\) is entry \(i\) of \(e\). We want the largest eigenvalue and associated positive eigenvector of (6.11).

After solving the principal eigenvalue problem, compute

(6.12)\[{\widetilde {\sf l}}_{ij} = \exp\left( - \tilde \eta \right) {\sf m}_{ij} \frac {e_j}{e_i}\]

and form the matrix \({\widetilde {\mathbb L}} = [{\widetilde {\sf l}}_{ij}]\). We have now constructed a matrix \({\widetilde {\mathbb L}}\) that behaves as a transition matrix for a different finite state Markov chain. Its entries are nonnegative, and

\[\sum_{j=1}^n {\widetilde {\sf l}}_{ij} = 1. \]

We may use this matrix for form increments \((X_t)'{\widetilde {\mathbb L}} X_{t+1}\) in a positive multiplicative martingale process \(\{{\widetilde L_t}\}\):

\[{\widetilde N}_{t+1} = (X_t)'{\widetilde {\mathbb L}} X_{t+1}.\]

To achieve a Theorem 6.1 representation of the multiplicative functional \(M_t\), use formula (6.12) for \({\widetilde{\sf m}}_{ij}\) to get \( {\sf m}_{ij} = \exp\left( \tilde \eta \right) {\widetilde {\sf m}}_{ij} \frac {{\sf e}_i}{{\sf e}_j}. \) This allows us to write (6.10) as

(6.13)\[{\frac {M_{t+1}}{M_t}} = \exp\left( \tilde \eta \right) \left[(X_t)'{\widetilde {\sf M}} X_{t+1}\right] \left( \frac {e \cdot X_t }{e \cdot X_{t+1}} \right) .\]

6.5. Stochastic stability

Recall that our characterization of a change of probability measure implied by the solution to a Perron-Frobenius problem only determines transition probabilities. Since the process is Markov, we may be able to find solutions for the initial distribution of \(X_0\) under which the process is stationary. When the eigenfunction problem has multiple solutions, it happens that there is a unique solution for which the process \(X\) is stochastically stable under the implied change of measure, namely, the solution associated with the minimum eigenvalue. See [Hansen and Scheinkman, 2009] and [Hansen, 2012] for a formal treatment of the this problem in a continuous-time Markov setting.

Definition 6.2

A process \(X\) is stochastically stable under a probability measure \({\widetilde {Pr}}\) if it is stationary and \(\lim_{j \rightarrow \infty} {\widetilde E} \left[h(X_j) \mid X_0 = x \right] = {\widetilde E} \left[ h(X_0) \right]\) for any Borel measurable \(h\) satisfying \({\widetilde E} \vert h(X_t) \vert < \infty\).

Stochastic stability under the change of measure opens the door to some revealing long-term approximation. Suppose that

(6.14)\[f > 0 \textrm{ and } 0 < {\widetilde {\mathbb E}}\left[{\frac {f(X_t)} {{\tilde e}(X_t)}}\right] < \infty.\]

Then

\[{\frac 1 j} \log {\mathbb M}^jf(x) = {\tilde \eta} + {\frac 1 j} \log {\widetilde {\mathbb E}} \left[ {\frac {f(X_j)} {{\tilde e}(X_j)}} \Big| X_0 = x \right] - {\frac 1 j} \log {\tilde e} (x)\]

Since \(X\) is stochastically stable under \({\widetilde P}r\),

\[\lim_{j \rightarrow \infty} {\frac 1 j} \log {\mathbb M}^jf(x) = {\tilde \eta} ,\]

Under the restriction, after adjusting for the growth decay in the semigroup, we obtain a more refined approximation:

\[\lim_{j \rightarrow \infty} \exp (- {\tilde \eta} j ) {\mathbb M}^j f(x) = \lim_{j \rightarrow \infty} {\widetilde {\mathbb E}} \left[ {\frac {f(X_j)} {{\tilde e}(X_j)}} \Big| X_0 = x \right] {\tilde e}(x) = {\widetilde E}\left[{\frac {f(X_t)} {{\tilde e}(X_t)}}\right] \tilde e(x).\]

where we assume that \({\widetilde {\mathbb E}}\left[{\frac {f(X_t)} {{\tilde e}(X_t)}}\right] < \infty\). Once we adjust for the impact of \({\tilde \eta}\), the limiting function is proportional to \({\tilde e}\). The function \(f\) determines only a scale factor \({\widetilde {\mathbb E}}\left[{\frac {f(X_t)} {{\tilde e}(X_t)}}\right] \tilde e(x)\).

It also turns out that stochastic stability is sufficient for the Perron-Frobenius eigenvalue problem to have a unique solution.

Theorem 6.2

Let \(M\) be a multiplicative functional. Suppose that \((\tilde \eta, \tilde e)\) solves eigenfunction problem and that under the change of measure \(\widetilde P\) implied by the associated martingale \(\widetilde M\) the stochastic process \(X\) is stationary and ergodic. Consider any other solution \((\eta^*, e^*)\) to eigenfunction problem with implied martingale \(\{ M_t^* \}\). Then

1. \(\eta^* \ge \tilde \eta\).

2. If \(X\) is stochastically stable under the change of measure \(Pr^*\) implied by the martingale \(M^*\), then \(\eta^* = \tilde \eta\), \(e^*\) is proportional to \(\tilde e\), and \(M^* = \widetilde M\) for all \(t=0,1,... \).

Proof

First we show that \(\eta^* \ge \tilde \eta\). Write:

\[ \mathbb{M}^t{e^*}(x) = \exp\left( \tilde \eta t \right) \widetilde{\mathbb E} \left( \left[ \frac { \tilde e(X_0)}{\tilde e(X_t) } \right] {e^*}(X_t) \Biggl| X_0=x \right) = \exp \left( \eta^* t \right) e^*(x) . \]

Thus,

\[ \widetilde{\mathbb E} \left( \left[ \frac { e^*(X_t)}{\tilde e(X_t) } \right] \Biggl| X_0=x \right) = \exp \left( \eta^* t - \tilde \eta t \right) \left[ \frac {e^*(x)}{\tilde e(x)} \right] . \]

If \(\tilde \eta > \eta^*\), then

\[ \lim_{t \rightarrow \infty} \widetilde{E} \left( \left[ \frac { e^*(X_t)}{\tilde e(X_t) } \right] \Biggl| X_0=x \right) = 0. \]

But this equality cannot be true because under \(\widetilde{Pr}\) \(X\) is stochatically stable and \(\frac {e^*}{\tilde e}\) is strictly positive. Therefore, \(\eta^* \ge {\tilde \eta}\)

Consider next the case in which \(\eta^* > \tilde \eta\). Write

\[ \frac {M_t}{ M_0} = \exp\left( \eta^* t \right) \left( \frac { M^*_t}{ M^*_0 } \right) \left( \frac {e^*(X_0)}{e^*(X_t) } \right), \]

which implies that

\[ \mathbb{M}^t\tilde e(x) = \exp\left( \eta^* t \right) E^* \left[ \left( \frac {e^*(X_0)}{e^*(X_t) } \right) \tilde e(X_t) \Biggl| X_0=x \right] = \exp \left( \tilde \eta t \right) \tilde e(x) . \]

Thus,

\[ {\mathbb E}^* \left( \left[ \frac { \tilde e(X_t)}{e^*(X_t) } \right] \Biggl| X_0=x \right) = \exp\left( \tilde \eta t - \eta^* t \right)\left[ \frac {\tilde e(x)}{e^*(x)}\right]. \]

Suppose that \(\tilde \eta < \eta^*\), then

\[ \lim_{t \rightarrow \infty} {\mathbb E}^* \left( \left[ \frac { \tilde e(X_t)}{e^*(X_t) } \right] \Biggl| X_0=x \right) = 0 , \]

so that \(X\) cannot be stochastically stable under the \(Pr^*\) measure.

Finally, suppose that \(\tilde \eta = \eta^*\) and that \(\frac {\tilde e(x)}{e^*(x)}\) is not constant. Then

\[ {\mathbb E}^* \left( \left[ \frac { \tilde e(X_t)}{e^*(X_t) } \right] \Biggl| X_0=x \right) = \frac {\tilde e(x)}{e^*(x)} \]

and \(X\) cannot be stochastically stable under the \(Pr^*\) measure.

We will apply results in a variety of ways in this chapter and in subsequent chapters. So far, we have shown how to construct a factorization of a multiplicative functional from an underlying stochastic model of the process. As we will see, it can help us understand better the valuation implications of stochastic equilibrium models. An alternative way to use these tools is look for direct empirical counterparts that open the door to testing or evaluating such models based directly on empirical evidence.

6.6. Inferences about permanent shocks

Empirical macroeconomics model and measure the impact of shock using macroeconomic variables measured in logarithms. For such models, the additive decomposition derived and analyzed in Chapter Processes with Markovian increments. [Alvarez and Jermann, 2005] suggest looking at asset pricing evidence to provide further evidence about this using multiplicative-type representations of the cumulative stochastic discount factor, but without the formalism developed in this chapter. We now explore this question.

We start with a factorization of a stochastic discount factor process as given in Theorem 6.1.

(6.15)\[\frac {S_t}{S_0} = \exp(t \eta^s) L_t^s \left[ \frac {e^s(X_0)}{e^s(X_t)} \right] \]

Taking logarithms, we form:

\[\log S_t - \log S_0 = t \eta^s + \log L_t^s + \log e^s(X_0) - \log e^s(X_t).\]

This looks like an additive decomposition of the type analyzed in Chapter Processes with Markovian increments, it is different. While \(L^s\) is a multiplicative martingale, \(\log L^s\) is typically a super martingale, but not a martingale. This leads us to write the additive decomposition as:

\[\log S_t - \log S_0 = t {\hat \eta}^s + {\widehat L}_t^s + {\hat e}^s(X_0) - {\hat e}^s(X_t)\]

where \({\widehat L}_t^s\) is an additive martingale. As [Hansen, 2012] argues, a weaker result does hold. If \(L^s\) is not degenerate (equal to one), then \({\widehat L}\) is not degenerate (equal to zero) and conversely. A prominent multiplicative martingale component implies a prominent role for permanent shocks in the underlying economic dynamics. With a formal probability model we have ability to go back and forth between the two representations as the results in this chapter and Chapter Processes with Markovian increments suggest. Example Example 6.6 gives an example of the distinction with explicit formulas linking the two representations.

6.7. Empirical counterparts to the factorization of stochastic discount factors

Consider again the stochastic discount factorization: (6.15). The date zero price of a long-term bond is:

\[{\mathbb E} \left( \frac {S_t}{S_0} \Biggl| {\mathfrak A}_0 \right) = \exp(\eta^s t) \widetilde {\mathbb E} \left[ \frac {1}{e^s(X_t)} \Biggl| X_0 \right] e^s(X_0). \]

Compute the corresponding yield by taking \(1/t\) times minus the logarithm:

\[- \eta^s - {\frac 1 t} \log \widetilde {\mathbb E} \left[ \frac {1}{e^s(X_t)} \Biggl| X_0 \right] + \frac 1 t \log e^s(X_0).\]

Provided that

\[\widetilde {\mathbb E} \left[ \frac {1}{e^s(X_t)} \Biggl|X_0 \right] < \infty,\]

the limiting yield on a discount bond is: \(- \eta^s.\)

Next consider the one-period holding period return on a \(t\) period discount bond:

\[\frac {\exp[\eta^s (t-1)] \widetilde {\mathbb E} \left[ \frac {1}{e^s(X_t)} \Biggl| X_1 \right] e^s(X_1)}{\exp(\eta^s t) \widetilde {\mathbb E} \left[ \frac {1}{e^s(X_t)} \Biggl| X_0 \right] e^s(X_0)}\]

Using stochastic stability and taking limits as \(t\) tends to \(\infty\) gives the limiting holding-period return:

(6.16)\[R_1^{\infty} \eqdef \exp(-\eta^s) \frac{ e^s(X_1)} {e^s(X_0)} . \]

An easy calculation shows that this satisfies the one-period pricing relation for a return:

\[{\mathbb E} \left[ \left(\frac {S_1}{S_0} \right) R_1^\infty \Biggl| {\mathfrak A}_0 \right] = {\widetilde {\mathbb E}} \left( \exp(\eta^s) \left[\frac{ e^s(X_0)} {e^s(X_1)} \right] \exp(-\eta^s) \left[\frac{ e^s(X_1)} {e^s(X_0)}\right] \Biggl| X_0 \right) = 1.\]

These long-horizon limits provide approximations to the eigenvalue for the stochastic discount factor and the ratio of the eigenfunctions. [Kazemi, 1992] observes in a model without a martingale component that the inverse of this holding-period return is the one-period stochastic discount factor. [Alvarez and Jermann, 2005] extends this insight by showing that the reciprocal reveals the component of one-period stochastic discount factor net of its martingale component. A subjective belief model could be a rationale for the martingale component of the cumulative stochastic discount factor process with stationary (net of a time trend) model of the macro economy.

In practice, we only have finite horizon bond data at our disposal so empirical implementation is base on the assumption the observed term structure data has a sufficiently long duration component to provide credible proxies.

6.8. Long-term risk-return tradeoff for cash flows

Following [Hansen and Scheinkman, 2009] and [Hansen et al., 2008], we consider the valuation of stochastic cash flows, \(G\), that are multiplicative functionals. These are cash flows are the more primitive inputs into both equities and bonds and contributors to even one-period asset pricing. Some recent research has explored direct measurement of the pricing of such cash flows.

We now study the long-term limits of such pricing. In addition to the stochastic discount process (6.15), form:

\[\log G_t - \log G_0 = t \eta^g + \log L_t^g + \log e^s(X_0) - \log e^s(X_t), \]

with a corresponding cash-flow return over horizon \(t\):

\[\frac {{G_t}}{ {\mathbb E} \left[ \left( \frac {S_t}{S_0} \right) G_ t \Biggl| {\mathfrak A}_0 \right] } = \frac {\frac {G_t}{G_0}} { {\mathbb E} \left( \frac {S_tG_t}{S_0G_0} \Biggl| X_0 \right)}. \]

Note that as a special case, the cash-flow return on a unit date \(t\) cash-flow is:

\[\frac 1 {{\mathbb E} \left( \frac {S_t}{S_0} \Biggl| X_0 \right)}\]

Define the proportional risk premium on the initial cash-flow return defined as:

(6.17)\[\frac 1 t \log {\mathbb E}\left( \frac {G_t}{G_0} \Biggl| X_0 \right) - \frac 1 t \log {\mathbb E} \left( \frac {S_tG_t}{S_0G_0} \Biggl| X_0 \right) + \frac 1 t \log {\mathbb E}\left( \frac {S_t}{S_0} \Biggl| X_0 \right)\]

where the third term is minus the logarithm of the riskless cash-flow return for horizon \(t\). We scale by \(1/t\) to adjust for the investment horizon.

Observe that the product \(SG\) is itself a multiplicative functional. Let \(\eta^{sg}\) denote its geometric growth component. Then from (6.17), the limiting cash-flow risk compensation is:

\[\eta^g + \eta^s - \eta^{gs}. \]

While this expression is reminiscent of a covariance measure, it is not literally in this mathematical form because we are working with proportional measures of risk compensation for positive payoffs.

Remark 6.2

An important special case of a cash flow is a cumulative return process, \(R\). For such a process: \(R_t/R_\tau\) for \(\tau < t\) is a \(t - \tau\) period return for any such \(t\) and \(\tau\). Normalize \(R_0 = 1\) and \(S_0 = 1\). For such a cash flow, \(SR\) is a multiplicative martingale implying that \(\eta^{sg} = 0\) and hence the limiting proportional risk premium is \(\eta^r + \eta^s.\) [Martin, 2012] studies cumulative returns and considers their tail behavior. Since \(SR\) is martingale bounded from below, it converges almost surely, and typically to zero. Given its date zero conditional expectation is one, this process necessary has a fat right tail for long horizons.

We also investigate the limiting behavior of one-period holding period returns. The empirical asset pricing literature has explored these returns starting with [van Binsbergen et al., 2012]. See [Golez and Jackwerth, 2024] for a recent update of this evidence. Using the factorization of \(SG\):

\[\log G_t - \log G_0 + \log S_t - \log S_0 = \eta^{sg} + \log L_t^{sg} + \log e^{gs}(X_0) - \log e^{gs}(X_t), \]

Since the we are constructing a multiplicative factorization, this construction is typically not the same as if we difference the logarithms of the individual factorizations of \(S\) and \(G\). We provide a characterization of the limiting one-period holding period return for the cash flow by imitating and extending our derivation of limiting holding-period return for riskless bond. This gives the following analogous formula to (6.16):

\[\exp( - \eta^{sg} )\left( \frac {G_1} {G_0} \right) \left[ \frac {e^{sg}(X_1) } {e^{sg} (X_0) } \right]. \]

While the eigenvalue and eigenfunction adjustments come from studying \(SG\) instead of \(S\), we also now inherit a stochastic growth term: \(G_1/G_0\). By multiplying this return by \(S_1/S_0\) we obtain \(N_1^{sg}\), which is the date one martingale increment for \(SG\). The one-period pricing relation for the cash-flow holding-period return follows immediately.

Finally, suppose that \(L^s = 1.\) In other words, the martingale component of the stochastic discount factor process is degenerate. Then \(SG\) inherits the martingale component of \(G\) implying that

\[\begin{align*} \eta^{sg} & = \eta^s + \eta^g\cr e^{sg}(x) & = e^s(x) e^g(x) \end{align*}\]

As a consequence, the long-term risk-return tradeoff is zero since proportional risk compensation in the limit is

\[\eta^s + \eta^g - \eta^{sg} = 0. \]

6.9. Recovering or bounding investor beliefs

We consider two very different approaches for drawing inferences about investor beliefs using the cumulative stochastic discount factorization.

6.9.1. Subjective beliefs in the absence of long-term risk

Suppose that we have direct data on the pricing of one-period contingent claims. Then the one-period operator, \({\mathbb M},\) could be inferred from data. Recall that we represent this operator using a baseline specification of the one-period transition probabilities. Rational expectations models take these transition probabilities to be same as those used by investors. Suppose instead that

• we endow investor with subjective beliefs that are possibly distinct from rational expectations;

• investors do think there are permanent shocks in the macroeconomy;

• investors do not have risk-based preferences that induce a multiplicative martingale in the cumulative stochastic discount factor process.

The third restriction is violated by recursive utility models of investor preference that we will analyze in a subsequent chapter. See discussions in [Alvarez and Jermann, 2005], [Hansen and Scheinkman, 2009], and [Borovička et al., 2016]. Under these circumstances, we could identify the \(L^s\) as the likelihood ratio for investor beliefs relative to the baseline probability distribution. Thus the implied martingale component in the cumulative stochastic discount factor identifies the subjective beliefs of investors. Using this change of measure, the limiting long-term risk compensations derived in the previous section are zero. These assumptions allow for what is called the “Ross recovery” of investor beliefs. This is entirely consistent with the formal analysis in [Ross, 2015], although Ross’s derives and motivates his result somewhat differently.

6.9.2. Bounding the martingale increment with limited asset market data

As an entirely different approach, suppose we maintain rational expectations by endowing investors with knowledge of the data generating process. With limited asset market data we cannot identify the martingale component to cumulative stochastic discount factor process without additional model restrictions. We can, however, obtain potentially valuable bounds on the martingale increment. In contrast to [Alvarez and Jermann, 2005], we use the increment as a device to represent conditional probabilities instead of a just random variable. We know that the implied martingale, as a stochastic process, has some unusual behavior while the implied probability measure can be well behaved.

There is a substantial literature on divergence measures for probability densities. Relative entropy, which we mentioned previously is one such example. More generally, consider a convex function \(\phi\) that is zero when evaluated at one. The function \(n\log n\) and \(-\log n\) are examples of such functions. An implication of Jensen’s inequality is that

\[{\mathbb E} \left[ \phi(N_1) \mid X_0\right] \ge 0,\]

and equal to zero when \(N_1\) is one provided that \(N_1\) is a multiplicative martingale increment (has conditional expectation one). This gives rise to a family of \(\phi\) divergences used to assess departures from baseline probabilities. Relative entropy, \(\phi(n) = n \log n\) is an example that is particularly tractable and has been used a variety of fields. The use of \(n \log n\) and \(- \log n\) have interpretations as expected log-likelihood ratios.

An approach to assessing the magnitude of \(N_1^L\) solves:

Minimum divergence Problem

\[\min_{N_1 \ge 0} {\mathbb E} \left[ \phi(N_1) \mid X_0\right] \]

subject to:

\[\begin{align} {\mathbb E} \left(N_1 \mid X_0 \right) & = 1 \cr {\mathbb E} \left(N_1 \left[\exp(\eta^s) \frac{ e^s(X_0)} {e^s(X_1)}\right] Y_1\Biggl| X_0 \right) & = Q_0 \end{align}\]

where \(Y_1\) is a vector of asset payoffs and \(Q_0\) is a vector of corresponding prices.

Recall that the term:

\[\left[\exp(\eta^s) \frac{ e^s(X_0)} {e^s(X_1)}\right] = \left(R_1^\infty \right)^{-1}\]

can be approximated by the reciprocal of the one-period holding-period return on a long-term bond.

This approach is an example of partial identification because the vector \(Y_1\) of asset payoffs may not be sufficient to reconstruct all of the potential one-period asset payoffs and prices. That is, because of data limitations, an econometrician may choose to use incomplete data on financial markets.

Remark 6.3

Applications often study the unconditional counterpart to this problem to avoid having to estimate conditional expectations. For such problems, conditioning can be brought in through the “back door” using the by scaling payoffs and prices with variables in the conditioning information set along the lines suggested by [Hansen and Singleton, 1982] and [Hansen and Richard, 1987]. See [Bakshi and Chabi-Yo, 2012] and [Bakshi et al., 2017] for some implementations along these lines.

Remark 6.4

[Alvarez and Jermann, 2005] use \(- {\mathbb E}\left( \log S_1 + \log S_0 \right) \) as the objective to be minimized. Notice that

\[\log N_1^s = \log S_1 - \log S_0 + [\eta^s + \log e^s(X_1) - \log e^s(X_0) ] \]

where the term in square brackets is the logarithm of the limiting holding-period bond return. Their objective is thus just the same as that in the minimum divergence problem with an additive translation. Rewrite the constraints as:

\[\begin{align} {\mathbb E} \left(\frac {S_1}{S_0} R_1^\infty \right) & = 1 \cr {\mathbb E} \left( \frac {S_1}{S_0} Y_1\right) & = Q_0. \end{align}\]

Thus we are left with an equivalent minimization problem in which the translation term is subtracted off to obtain the bound of interest.

Applied researchers have omitted the first constraint, which weakens the bound. Also [Chen et al., 2024] isolate a potentially problematic aspect of monotone decreasing divergences because they may fail to detect certain limiting forms of deviations from baseline probabilities.

Remark 6.5

[Chen et al., 2020] propose extensions of the one-period divergence measures to multi-period counterparts that remain tractable and revealing. Their method for accommodating conditioning information for bounding such divergences has a direct extension to the problem considered here.

Remark 6.6

Notice that if the martingale component of the stochastic discount factor is identically one, then a testable implication is:

\[{\mathbb E} \left[ \left(R_t^\infty\right)^{-1} Y_1 \mid X_0 \right] = Q_0 . \]