Some Multiplicative Functionals

Daisuke Oyama, Thomas J. Sargent and John Stachurski

Plan of the notebook

In other quant-econ lectures ("Markov Asset Pricing" and "The Lucas Asset Pricing Model"), we have studied the celebrated Lucas asset pricing model (Lucas (1978)) that is cast in a setting in which the key objects of the theory, namely, a stochastic discount factor process, an aggregate consumption process, and an asset payout process, are all taken to be stationary.

In this notebook, we shall learn about some tools that allow us to extend asset pricing models to settings in which neither the stochastic discount factor process nor the asset's payout process is a stationary process. The key tool is the class of multiplicative functionals from the stochastic process literature that Hansen and Scheinkman (2009) have adapted so that they can be applied to asset pricing and other interesting macroeconomic problems.

In this notebook, we confine ourselves to studying a special type of multiplicative functional, namely, multiplicative functionals driven by finite state Markov chains. We'll learn about some of their properties and applications. Among other things, we'll obtain Hansen and Scheinkman's more general multiplicative decomposition of our particular type of multiplicative functional into the following three primitive types of multiplicative functions:

• a nonstochastic process displaying deterministic exponential growth;

• a multiplicative martingale or likelihood ratio process; and

• a stationary stochastic process that is the exponential of another stationary process.

The first two of these primitive types are nonstationary while the third is stationary. The first is nonstochastic, while the second and third are stochastic.

After taking a look at the behavior of these three primitive components, we'll apply this structure to model

• a stochastically, exponentially declining stochastic discount factor process;

• a stochastically, exponentially growing or declining asset payout or dividend process;

• the prices of claims to exponentially growing or declining payout processes; and

• a theory of the term structure of interest rates.

We begin by describing a basic setting that we'll use in several applications later in this notebook.

Multiplicative functional driven by a finite state Markov chain

Let $S$ be the integers $\{0, \ldots, n-1\}$. Because we study stochastic processes taking values in $S$, elements of $S$ will be denoted by symbols such as $x, y$ instead of $i, j$. Also, to avoid double subscripts, for a vector $h \in {\mathbb R}^n$ we will write $h(x)$ instead of $h_x$ for the value at index $x$. (In fact $h$ can also be understood as a function $h \colon S \to {\mathbb R}$. However, in expressions involving matrix algebra we always regard it as a column vector. Similarly, $S$ can be any finite set but in what follows we identify it with $\{0, \ldots, n-1\}$.)

Matrices are represented by symbols such as ${\mathbf P}$ and ${\mathbf Q}$. Analogous to the vector case, the $(x,y)$-th element of matrix ${\mathbf Q}$ is written ${\mathbf Q}(x, y)$ rather than ${\mathbf Q}_{x y}$. A nonnegative $n \times n$ matrix ${\mathbf Q}$ is called irreducible if, for any $(x, y) \in S \times S$, there exists an integer $m$ such that ${\mathbf Q}^m(x, y) > 0$. It is called primitive if there exists an integer $m$ such that ${\mathbf Q}^m(x, y) > 0$ for all $(x, y) \in S \times S$. A positive integer $d$ is the period of $x \in S$ if it is the greatest common divisor of all $m$'s such that ${\mathbf Q}^m(x, x) > 0$. If ${\mathbf Q}$ is irreducible, then all $x$'s in $S$ have the same period, which is called the period of the matrix ${\mathbf Q}$. ${\mathbf Q}$ is called aperiodic if its period is one. A nonnegative matrix is irreducible and aperiodic if and only if it is primitive.

Let ${\mathbf P}$ be a stochastic $n \times n$ matrix and let $\{X_t\}$ be a Markov process with transition probabilities ${\mathbf P}$. That is, $\{X_t\}$ is a Markov process on $S$ satisfying

$$ \mathbb P [ X_{t+1} = y \mid {\mathcal F}_t] = {\mathbf P}(X_t, y) $$

for all $y \in S$. Here $\{{\mathcal F}_t\}$ is the natural filtration generated by $\{X_t\}$ and the equality holds almost surely on an underlying probability space $(\Omega, {\mathcal F}, \mathbb P)$.

A martingale with respect to $\{{\mathcal F}_t\}$ is a real-valued stochastic process on $(\Omega, {\mathcal F}, \mathbb P)$ satisfying $E[|M_t|] < \infty$ and $E[M_{t+1} \mid {\mathcal F}_t] = M_t$ for all $t$. A multiplicative functional generated by $\{X_t\}$ is a real-valued stochastic process $\{M_t\}$ satisfying $M_0 > 0$ and

$$ \frac{M_{t+1}}{M_t} = {\mathbf M}(X_t, X_{t+1}) $$

for some strictly positive $n \times n$ matrix ${\mathbf M}$. If, in addition,

$$ E[ {\mathbf M}(X_t, X_{t+1}) \mid {\mathcal F}_t] = 1, $$

then $\{M_t\}$ is clearly a martingale. Given its construction as a product of factors ${\mathbf M}(X_t, X_{t+1})$, it is sometimes called a multiplicative martingale.

If we write

$$ \ln M_{t+1} - \ln M_t = {\mathbf G}(X_t, X_{t+1}) $$

where

$$ {\mathbf G}(X_t, X_{t+1}) := \ln {\mathbf M}(X_t, X_{t+1}), $$

then ${\mathbf G}(x, y)$ can be interpreted as the growth rate of $\{M_t\}$ at state pair $(x, y)$.

A likelihood ratio process is a multiplicative martingale $\{M_t\}$ with initial condition $M_0 = 1$. From this initial condition and the martingale property it is easy to show that

$$ E[M_t] = E[M_t \mid {\mathcal F}_0] = 1 $$

for all $t$.

Martingale decomposition

Let ${\mathbf P}$ be a stochastic matrix, let ${\mathbf M}$ be a positive $n \times n$ matrix and let $\{M_t\}$ be the multiplicative functional defined above. Assume that ${\mathbf P}$ is irreducible. Let $\widetilde {\mathbf P}$ be defined by

$$ \widetilde {\mathbf P} (x, y) = {\mathbf M}(x, y) {\mathbf P}(x, y) \qquad ((x, y) \in S \times S). $$

Using the assumptions that ${\mathbf P}$ is irreducible and ${\mathbf M}$ is positive, it can be shown that $\widetilde {\mathbf P}$ is also irreducible.

By the Perron-Frobenius theorem, there exists for $\widetilde {\mathbf P}$ a unique eigenpair $(\lambda, e) \in {\mathbb R} \times {\mathbb R}^n$ such that $\lambda$ and all elements of $e$ are strictly positive. Letting $\eta := \log \lambda$, we have

$$ \widetilde {\mathbf P} e = \exp(\eta) e. $$

Now define $n \times n$ matrix $\widetilde {\mathbf M}$ by

$$ \widetilde {\mathbf M}(x, y) := \exp(- \eta) {\mathbf M}(x, y) \frac{e(y)}{e(x)}. $$

Note that $\widetilde {\mathbf M}$ is also strictly positive. By construction, for each $x \in S$ we have

\begin{align*} \sum_{y \in S} \widetilde {\mathbf M}(x, y) {\mathbf P}(x, y) & = \sum_{y \in S} \exp(- \eta) {\mathbf M}(x, y) \frac{e(y)}{e(x)} {\mathbf P}(x, y) \\ & = \exp(- \eta) \frac{1}{e(x)} \sum_{y \in S} \widetilde {\mathbf P}(x, y) e(y) \\ & = \exp(- \eta) \frac{1}{e(x)} \widetilde {\mathbf P} e(x) = 1. \end{align*}

Now let $\{\widetilde M_t\}$ be the multiplicative functional defined by

$$ \frac{\widetilde M_{t+1}}{\widetilde M_t} = \widetilde {\mathbf M}(X_t, X_{t+1}) \quad \text{and} \quad \widetilde M_0 = 1. $$

In view of our proceeding calculations, we have

$$ E \left[ \frac{\widetilde M_{t+1}}{\widetilde M_t} \Bigm| {\mathcal F}_t \right] = E[ \widetilde {\mathbf M}(X_t, X_{t+1}) \mid {\mathcal F}_t] = \sum_{y \in S} \widetilde {\mathbf M}(X_t, y) {\mathbf P}(X_t, y) = 1. $$

Hence $\{\widetilde M_t\}$ is a likelihood ratio process.

By reversing the construction of $\widetilde {\mathbf M}$ given above, we can write

$$ {\mathbf M}(x, y) = \exp( \eta) \widetilde {\mathbf M}(x, y) \frac{e(x)}{e(y)} $$

and hence

$$ \frac{M_{t+1}}{M_t} = \exp( \eta) \frac{e(X_t)}{e(X_{t+1})} \frac{\widetilde M_{t+1}}{\widetilde M_t} . $$

In this equation we have decomposed the original multiplicative functional into the product of

1. a nonstochastic component $\exp( \eta)$,
2. a stationary sequence $e(X_t)/e(X_{t+1})$, and
3. the factors $\widetilde M_{t+1}/\widetilde M_t$ of a likelihood ratio process.

Simulation strategy

Let $x_t$ be the index of the Markov state at time $t$ and let $\{ x_0, x_1, \ldots, x_T\}$ be a simulation of the Markov process for $\{X_t\}$.

We can use the formulas above easily to generate simulations of the multiplicative functional $M_t$ and of the positive multiplicative martingale $\widetilde M_t$.

Forecasting formulas

Let $\{M_t\}$ be the multiplicative functional described above with transition matrix ${\mathbf P}$ and matrix ${\mathbf M}$ defining the multiplicative increments. We can use $\widetilde {\mathbf P}$ to forecast future observations of $\{M_t\}$. In particular, we have the relation

$$ E[ M_{t+j} \mid X_t = x] = M_t \sum_{y \in S} \widetilde {\mathbf P}^j(x, y) \qquad (x \in S). $$

This follows from the definition of $\{M_t\}$, which allows us to write

$$ M_{t+j} = M_t {\mathbf M}(X_t, X_{t+1}) \cdots {\mathbf M}(X_{t+j-1}, X_{t+j}). $$

Taking expectations and conditioning on $X_t = x$ gives

\begin{align*} E[ M_{t+j} \mid X_t = x] & = \sum_{(x_1, \ldots, x_j)} M_t {\mathbf M}(x, x_1) \cdots {\mathbf M}(x_{j-1}, x_j) {\mathbf P}(x, x_1) \cdots {\mathbf P}(x_{j-1}, x_j) \\ & = \sum_{(x_1, \ldots, x_j)} M_t \widetilde {\mathbf P}(x, x_1) \cdots \widetilde {\mathbf P}(x_{j-1}, x_j) \\ & = M_t \sum_{y \in S} \widetilde {\mathbf P}^j(x, y) . \end{align*}

Implementation

The MultFunctionalFiniteMarkov class implements multiplicative functionals driven by finite state Markov chains.

Here we briefly demonstrate how to use it.

Clearly, this Markov chain is irreducible:

Create a MultFunctionalFiniteMarkov instance:

The dominant eigenvalue, denoted $\exp(\eta)$ above, of $\widetilde P$ is

The value $\eta$ is

The (normalized) dominant eigenvector $e$ of $\widetilde P$ is

Let us simulate our MultFunctionalFiniteMarkov:

The simulation results are contained in res.

Let's check that M and M_tilde satisfy the identity from their definition (up to numerical errors).

Likelihood ratio processes

A likelihood ratio process is a multiplicative martingale with mean $1$.

A multiplicative martingale process $\{\widetilde M_t \}_{t=0}^\infty$ that starts from $\widetilde M_0 = 1$ is a likelihood ratio process.

Evidently, a likelihood ratio process satisfies

$$ E [\widetilde M_t \mid {\mathfrak F}_0] = 1 .$$

Hansen and Sargent (2017) point out that likelihood ratio processes have the following peculiar property:

• Although $E{\widetilde M}_{j} = 1$ for each $j$, $\{{\widetilde M}_{j} : j=1,2,... \}$ converges almost surely to zero.

The following graph, and also one at the end of this notebook, illustrate the peculiar property by reporting simulations of many sample paths of a $\{\widetilde M_t \}_{t=0}^\infty$ process.

We revisit the peculiar sample path property at the end of this notebook.

Stochastic discount factor and exponentially changing asset payouts

Define a matrix ${\sf S}$ whose $(x, y)$th element is ${\sf S}(x,y) = \exp(G_S(x,y))$, where $G_S(x,y)$ is a stochastic discount rate for moving from state $x$ at time $t$ to state $y$ at time $t+1$.

A stochastic discount factor process $\{S_t\}_{t=0}^\infty$ is governed by the multiplicative functional:

$$ {\frac {S_{t+1}}{S_t}} = \exp[ G_S(X_t, X_{t+1} ) ] = {\sf S}(X_t, X_{t+1}). $$

Define a matrix ${\sf D}$ whose $(x,y)$th element is ${\sf D}(x,y) = \exp(G_d(x,y))$.

A non-negative payout or dividend process $\{d_t\}_{t=0}^\infty$ is governed by the multiplicative functional:

$$ {\frac {d_{t+1}}{d_t}} = \exp\left[ G_d(X_t,X_{t+1}) \right] = {\sf D}(X_t, X_{t+1}). $$

Let $p_t$ be the price at the beginning of period $t$ of a claim to the stochastically growing or shrinking stream of payouts $\{d_{t+j}\}_{j=0}^\infty$.

It satisfies

$$ p_t = E\left[\frac{S_{t+1}}{S_t} (d_t + p_{t+1}) \Bigm| {\mathfrak F}_t\right] , $$

or

$$ \frac{p_t}{d_t} = E\left[\frac{S_{t+1}}{S_t} \left(1 + \frac{d_{t+1}}{d_t} \frac{p_{t+1}}{d_{t+1}}\right) \Bigm| {\mathfrak F}_t\right] , $$

where the time $t$ information set ${\mathfrak F}_t$ includes $X_t, S_t, d_t$.

Guessing that the price-dividend ratio $\frac{p_t}{d_t}$ is a function of the Markov state $X_t$ only, and letting it equal $v(x)$ when $X_t = x$, write the preceding equation as

$$ v(x) = \sum_{x \in S} P(x,y) \left[ {\sf S}(x,y) \mathbf{1} + {\sf S}(x,y) {\sf D}(x,y) v(y) \right] $$

or

$$ v = c + \widetilde{P} v , $$

where $c = \widehat{P} \mathbf{1}$ is by construction a nonnegative vector and we have defined the nonnegative matrices $\widetilde{P} \in \mathbb{R}^{n \times n}$ and $\widehat{P} \in \mathbb{R}^{n \times n}$ by

$$ \begin{aligned} \widetilde{P}(x,y) &= P(x,y) {\sf S}(x,y) {\sf D}(x,y), \\ \widehat{P}(x,y) &= P(x,y) {\sf S}(x,y). \end{aligned} $$

The equation $v = \widetilde{P} v + c$ has a nonnegative solution for any nonnegative vector $c$ if and only if all the eigenvalues of $\widetilde{P}$ are smaller than $1$ in modulus.

A sufficient condition for existence of a nonnegative solution is that all the column sums, or all the row sums, of $\widetilde{P}$ are less than one, which holds when $G_S + G_d \ll 0$. This condition describes a sense in which discounting counteracts growth in dividends.

Given a solution $v$, the price-dividend ratio is a stationary process that is a fixed function of the Markov state:

$$ \frac{p_t}{d_t} = v(x) \text{ when $X_t = x$}. $$

Meanwhile, both the asset price process and the dividend process are multiplicative functionals that experience either multiplicative growth or decay.

Implementation

The AssetPricingMultFiniteMarkov class implements the asset pricing model with the specification of the stochastic discount factor process described above.

Below is an example of how to use the class.

Please note that the stochastic discount rate matrix $G_S$ and the payout growth rate matrix $G_d$ are specified independently.

In the Lucas asset pricing model to be described below, the matrix $G_S$ is a function of the payoff growth rate matrix $G_d$ and another parameter $\gamma$ that is a coefficient of relative risk aversion in the utility function of a representative consumer, as well as the discount rate $\delta$.

(1) Display the $\widetilde M$ matrices for $S_t$ and $d_t$.

(2) Plot sample paths of $S_t$ and $d_t$.

(2) Print $v$.

(3) Plot sample paths of $p_t$ and $d_t$.

(5) Experiment with a different $G_S$ matrix.

Lucas asset pricing model with growth

As an example of our model of a stochastic discount factor and payout process, we'll adapt a version of the famous Lucas (1978) asset pricing model to have an exponentially growing aggregate consumption endowment. We'll use CRRA utility $u(c) = c^{1-\gamma}/(1-\gamma)$, a common specification in applications of the Lucas model.

So now we let $d_t = C_t$, aggregate consumption, and we let

$$\frac{S_{t+1}}{S_t} = \exp(-\delta) \left(\frac{C_{t+1}}{C_t} \right)^{-\gamma} ,$$

where $\delta > 0$ is a rate of time preference and $\gamma > 0$ is a coefficient of relative risk aversion.

To obtain this special case of our model, we set

$$ {\sf S}(x, y) = \exp(-\delta) {\sf D}(x, y)^{-\gamma}, $$

where we now interpret ${\sf D}(x, y)$ as the multiplicative rate of growth of the level of aggregate consumption between $t$ and $t+1$ when $X_t = x$ and $X_{t+1} = y$.

Term structure of interest rates

When the Markov state $X_t = x$ at time $t$, the price of a risk-free zero-coupon bond paying one unit of consumption at time $t+j$ is

$$ p_{j,t} = E \left[ \frac{S_{t+j}}{S_t} \Bigm| X_t = x \right]. $$

Let the matrix $\widehat{P}$ be given by $\widehat{P}(x, y) = P(x, y) {\sf S}(x, y)$ and apply the above forecasting formula to deduce

$$ p_{j,t} = \left( \sum_{y \in S} \widehat P^j(x, y) \right). $$

The yield $R_{jt}$ on a $j$ period risk-free bond satisfies

$$ p_{jt} = \exp(-j R_{jt}) $$

or

$$ R_{jt} = -\frac{\log(p_{jt})}{j}. $$

For a given $t$,

$$ \begin{bmatrix} R_{1t} & R_{2t} & \cdots & R_{Jt} \end{bmatrix} $$

is the term structure of interest rates on risk-free zero-coupon bonds.

Simulating the Lucas asset pricing model

Write $y$ for the process of quarterly per capita consumption growth with mean $\mu_C$. In the following example, we assume that $y - \mu_C$ follows a discretized version of an AR(1) process (while independent of the Markov state), where the discrete approximation is derived by the routine tauchen from quantecon.markov.

Create a LucasTreeFiniteMarkov instance:

Simulate the model:

Plotting the term structure of interest rates

The term structure of interest rates R is a sequence (of length J) of vectors (of length n each). Instead of plotting the whole R, we plot the sequences for the "low", "middle", and "high" states.

Here we define those states as follows. The vector $(p_{jt}|X_t = x)_{x \in S}$, if appropriately rescaled, converges as $j \to \infty$ to an eigenvector of $\widehat P$ that corresponds to the dominant eigenvalue, which equals mf.e times some constant. Thus call the states that correspond to the smallest, largest, and middle values of mf.e the high, low, and middle states.

Another class of examples

Let the elements of ${\sf D}$ (i.e., the multiplicative growth rates of the dividend or consumption process) be, for example, $$ {\sf D} = \begin{bmatrix} .95 & .975 & 1 \cr .975 & 1 & 1.025 \cr 1 & 1.025 & 1.05 \end{bmatrix}.$$

Here the realized growth rate depends on both $X_t$ and $X_{t+1}$ -- i.e., the value of the state last period (i) and this period (j).

Here we have imposed symmetry to save parameters, but of course there is no reason to do that.

We can combine this specification with various specifications of $P$ matrices e.g., an "i.i.d." state evolution process would be represented with $P$ in which all rows are identical. Even that simple specification can some interesting outcomes with the above ${\sf D}$.

We'll try this little $3 \times 3$ example with a Lucas model below.

But first a word of caution.

We have to choose values for the consumption growth rate matrix $G_C$ and the transition matrix $P$ so that pertinent eigenvalues are smaller than one in modulus.

This check is implemented in the code.

The peculiar sample path property revisited

Consider again the multiplicative martingale that associated with the $5$ state Lucas model studied earlier.

Remember that by construction, this is a likelihood ratio process.

Here we'll simulate a number of paths and build up histograms of $\widetilde M_t$ at various values of $t$.

These histograms should help us understand what is going on to generate the peculiar property mentioned above.

As $t \rightarrow +\infty$, notice that

• more and more probability mass piles up near zero, $\ldots$ but

• a longer and longer thin right tail emerges.

Observe that the sample mean in each panel, denoted by the dotted line, is very close to $1$.

References

• Lars Peter Hansen and Thomas J. Sargent (2017), Risk, Uncertainty, and Value, Princeton University Press, forthcoming.

• Lars Peter Hansen and Jose A. Scheinkman (2009), "Long-Term Risk: An Operator Approach," Econometrica, 77(1), 177-234.

• Robert E. Lucas, Jr. (1978), "Asset Prices in an Exchange Economy," Econometrica 46, 1429-1445.