Perturbing Multiplicative Functionals

9. Perturbing Multiplicative Functionals#

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Authors: Jaroslav Borovicka (NYU), Lars Peter Hansen (University of Chicago), and Thomas J. Sargent (NYU)

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../_images/BocconiNobelHansen.jpg

Artist’s drawing inspired by a talk entitled “Shock Elasticities and Impulse Responses.”

“The manner in which risk operates upon time preference will differ, among other things, according to the periods in the future to which the risk applies.” - Irving Fisher (Theory of Interest (1930))

9.1. Introduction#

Local methods open the door to intertemporal characterizations of asset valuation. Such characterizations have direct links to first-order conditions of investors. While macroeconomists often solve models and analyze the implied impulse response using time series characterizations expressed in terms of logarithms, for asset valuation with compensation for uncertainty exposure, it is imperative to work with levels instead of logarithms. Given the presence of stochastic growth contributions, we are led to explore perturbations of multiplicative functionals. We introduce multiplicative perturbations which in turn lead naturally to the use of elasticities as a way to represent intertemporal compensations. This work builds on insights from [Hansen and Scheinkman, 2012], [Borovička and Hansen, 2014], [Borovička et al., 2014], and [Borovička and Hansen, 2016]. This chapter focuses exclusively on discrete-time specifications. [Borovička et al., 2014] and [Hansen et al., 2024] discuss continuous-time counterparts.

9.2. An elasticity calculation#

We start by considering a family of positive random variables, \(N_1({\sf r})\) for \({\sf r} \ge 0\) with unit conditional expectations and with a limit \(N_1(0) = 1\). We use this family to depict date one perturbations. Let \(M\) be a multiplicative functional, which could be a stochastic discount factor process or a stochastic growth process. Compute:

\[{\mathbb E} \left[\left(\frac {M_t}{M_0} \right) N_1({\sf r}) \vert X_0 \right].\]

We use two interpretations of this computation:

  • \(N_1({\sf r})\) induces a date one change in distribution;

  • \(N_1({\sf r})\) induces a change in the date one exposure to uncertainty.

Both will be of interest to us going forward. The first one allows us to construct a type of expected impulse response function where we change the initial distribution of a shock. The second defines a family of alternative cash flows for which we may deduce compensations. We introduce the positive scalar \({\sf r}\) in order that we can perform local characterizations in terms of derivatives of the form:

(9.1)#\[\epsilon^m(x,t) \eqdef \frac {d} {d{\sf r}} \log {\mathbb E} \left[ \left(\frac {M_t}{M_0} \right) N_1({\sf r}) \mid X_0 = x\right] \vert_{{\sf r} = 0} = \frac { \frac {d} {d{\sf r}} {\mathbb E} \left[ \left(\frac {M_t}{M_0} \right) N_1({\sf r}) \mid X_0 = x \right] \vert_{{\sf r} = 0}}{ {\mathbb E} \left[\left(\frac {M_t}{M_0} \right) \mid X_0 \right] } \]

which is in the form of an elasticity. The denominator on the right side offsets growth in the \(M\) process. Note also that

\[\frac {d} {d{\sf r}} \log N_1({\sf r}) \vert_{{\sf r} = 0} = \frac {d} {d{\sf r}} N_1({\sf r})\vert_{{\sf r} = 0}\]

since \(N_1(0) = 1\). Given this, we refer to computation (9.1) as an elasticity.

9.3. An important special case#

Write the evolution of the multiplicative functional as:

\[\log M_{t+1} - \log M_t = \kappa(X_t, W_{t+1} ) = \kappa_1(X_t) + \kappa_2(X_t) \cdot W_{t+1} \]

and the proportional perturbation expressed in logarithms as

\[\log N_1({\sf r}) = {\sf r} \pi(X_0) \cdot W_{1} - {\frac 1 2} | \pi(X_0)|^2 ({\sf r})^2\]

where \(W_1\) is a multivariate standard normally distributed random vector that is independent of \(X_0\). Clearly, \(N_1(0) = 1,\) and by properties of the log-normal distribution:

\[{\mathbb E} \left[ N_1({\sf r}) \mid X_0 = x \right] = 1. \]

The vector \(\pi\) gives a possibly state-dependent way to select among the different possible shocks.

Provided that we can differentiate inside the expectation operator, we find that

(9.2)#\[\epsilon^m(x,t) = \pi(x) \cdot \frac {{\mathbb E} \left[\left(\frac {M_t}{M_0} \right) W_1 \mid X_0 \right] }{{\mathbb E} \left[\left(\frac {M_t}{M_0} \right) \mid X_0 \right] }\]

Under the first interpretation of this family of perturbations, we change the distribution of \(W_1\) from being a multivariate, standard normal to a normal with mean \({\sf r} \pi(X_0)\) and an identity as the covariance matrix. That is, we perturb the shock distribution by including a nonzero mean for the date one shock vector. Under the second interpretation, we change the evolution of \(\log M\) by setting:

\[\log M_1 - \log M_0 = \kappa_1(X_0) + \kappa_2(X_0) \cdot W_1 + {\sf r} \pi(X_0) \cdot W_1 - {\frac 1 2} | \pi(X_0)|^2 ({\sf r})^2. \]

With this construction, we have changed the exposure to \(W_1\) of the multiplicative functional, an impact that persists over time. Since we look at limits as \({\sf r}\) declines to zero, the third term becomes dominated by the second.

We now investigate what happens at the one-period horizon and what happens when the horizon becomes arbitrarily long. For \(t=1\), note that \(M_1/M_0\) is conditional log normal with a conditional expectation:

\[\exp\left[ \kappa_1(x) - \frac 1 2 {\sf r}^2\vert \pi(x)\vert^2 + \frac 1 2 \left\vert \kappa_2(x) + {\sf r} \pi(x) \right\vert^2 \right]\]

Differentiating the logarithm with respect to \({\sf r}\) and evaluating this derivative at zero gives:

\[\epsilon^m(x,1) = \pi(x) \cdot \kappa_2(x) .\]

To study the long-horizon counterpart, it is revealing to use the martingale factorization to represent the shock elasticities. Recall that

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

where \({\widetilde L}\) is a multiplicative martingale. With this factorization, we use the Law of Iterated Expectations to write

(9.3)#\[\begin{align} \frac {{\mathbb E} \left[ \left(\frac {M_t}{M_0} \right) \mid {\mathfrak A}_1\right]}{{\mathbb E} \left[ \left(\frac {M_t}{M_0} \right) \mid {\mathfrak A}_0\right]} & = \frac{{\mathbb E} \left[ \left(\frac {M_t}{M_1} \right) \mid {\mathfrak A}_1 \right] \left( \frac {M_1}{M_0}\right)} {{\mathbb E} \left[ \left(\frac {M_t}{M_0} \right) \mid {\mathfrak A}_0 \right]} \cr & = \left[\frac {{\mathbb E} \left( \frac{{\widetilde L}_t}{{\widetilde L}_1} \left[ \frac 1 {{\tilde e}(X_t)}\right] \mid {\mathfrak A}_1 \right) } {{\mathbb E} \left( {\widetilde L}_t \left[ \frac 1 {{\tilde e}(X_t)} \right] \mid {\mathfrak A}_0 \right)}\right]{\widetilde L}_1 \end{align}\]

The random variable on the left is positive and has expectation one conditioned on \({\mathfrak A}_0\). Substituting this calculation into formula (9.2) and applying the Law of Iterated Expectations gives:

(9.4)#\[\epsilon^m(x,t) = \pi(x) \cdot {\mathbb E} \left[ \left(\frac {{\mathbb E} \left[ \left(\frac{{\widetilde L}_t}{{\widetilde L}_1}\right) \left[ \frac 1 {{\tilde e}(X_t)} \right] \mid X_1\right]} {{\mathbb E} \left( {\widetilde L}_t \left[ \frac 1 {{\tilde e}(X_t)} \right] \mid X_0 \right)}\right) {\widetilde L}_1 W_1 \ \Biggl| X_0 = x \right] \]

Under stochastic stability (Definition 8.2 of Chapter 8) of the change of probability measure induced by the martingale \({\widetilde L}\), the random variable on the right converges to \({\widetilde L}_1\) as \(t\) tends to \(\infty\). These calculations suggest defining the limiting elasticity as:

\[\epsilon^m(x,\infty) \eqdef \pi(x) \cdot {\mathbb E} \left( {\widetilde L}_1 W_1 \mid X_0 = x \right) .\]

Formally, the convergence to this limit requires more than point-wise or almost sure convergence of relative densities in (9.3) for the conditional expectations to converge, but these conditions are often satisfied in applications.

Example 9.1

We now revisit Example 8.6. That is, consider \(M = \exp(Y)\) constructed with a stationary \(X\) process and an additive \(Y\) process described by the VAR

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

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 \({\mathbb I}\). Compute

(9.5)#\[\begin{align} {\mathbb E} \left(Y_t - Y_0 \mid {\mathfrak A}_1 \right) & = t \nu + {\mathbb D}\sum_{j=0}^{t-2} {\mathbb A}^j X_1 + {\mathbb D} X_0 + {\mathbb F} W_1 \cr & = t \nu + {\mathbb D} \sum_{j=0}^{t-1} {\mathbb A}^{j} X_0 + \left[ {\mathbb D}\sum_{j=0}^{t-2} {\mathbb A}^{j} {\mathbb B} + {\mathbb F} \right] W_1 . \end{align}\]

where the second line follows by substituting for the dynamic evolution of \(X_1\). Applying the Law of Iterated Expectations:

(9.6)#\[{\mathbb E} \left(Y_t - Y_0 \mid {\mathfrak A}_0 \right) = t \nu + {\mathbb D} \sum_{j=0}^{t-1} {\mathbb A}^{j} X_0.\]

Subtracting (9.6) from (9.5) gives the forecast error:

\[\left[ {\mathbb D}\sum_{j=0}^{t-2} {\mathbb A}^{j} {\mathbb B} + {\mathbb F}\right] W_1\]

With this finding and the properties of the log-normal distribution:

\[\frac {{\mathbb E}\left[\exp \left(Y_t - Y_0\right) \mid {\mathfrak A}_1 \right] } {{\mathbb E}\left[\exp \left(Y_t - Y_0\right) \mid {\mathfrak A}_0 \right]} = \exp\left( \left[ {\mathbb D} \sum_{j=0}^{t-2} {\mathbb A}^{j} {\mathbb B} + {\mathbb F} \right] W_1 - {\frac 1 2} \left\vert{\mathbb D}\sum_{j=0}^{t-2} {\mathbb A}^{j} {\mathbb B} + {\mathbb F} \right\vert^2 \right)\]

This random variable induces a change in distribution for the standard normally distributed random vector \(W_1\) by endowing it with a conditional mean:

\[ {\mathbb B}' \sum_{j=0}^{t-2} \left({{\mathbb A}'}\right)^{j} {\mathbb D} ' + {\mathbb F}' ,\]

and the identity as the covariance matrix. Thus

\[\epsilon^m(x,t) = \pi \cdot \left[ {\mathbb B}' \sum_{j=0}^{t-2} \left({{\mathbb A}'}\right)^{j} {\mathbb D}' + {\mathbb F}' \right]\]

This coincides with the horizon-dependent impulse responses from standard analyses of a vector-autoregressive system where \(\pi\) selects the shock of interest. These elasticities do not depend on the initial state vector.

This special case featured shocks with normal distributions and abstracted from state dependence in the conditional volatilities. The methods described in the previous section are much more generally applicable.

9.4. Multiplicative martingale#

When \(M\) is a martingale (\(M = {\widetilde L}\)), it follows from the right side of (9.1) and the Law of Iterated Expectations that

\[\epsilon^m(x,t) = \frac { \frac {d} {d{\sf r}} {\mathbb E} \left[\left(\frac {M_1}{M_0} \right) N_1({\sf r}) \mid X_0 = x \right] \vert_{{\sf r} = 0}}{ {\mathbb E} \left[\left(\frac {M_1}{M_0} \right) \mid X_0 \right] },\]

and is thus constant as a function of \(t \ge 1\). For such a process, we sometimes find a second type of elasticity to be of interest. Suppose we perturb the process at date \(t\) instead of date one, giving rise to:

(9.7)#\[\varepsilon^m(x,t) \eqdef \frac {d} {d{\sf r}} \log {\mathbb E} \left[\left(\frac {M_t}{M_0} \right) N_t({\sf r}) \mid X_0 = x\right] \vert_{{\sf r} = 0} = \frac {d} {d{\sf r}} {\mathbb E} \left[\left(\frac {M_t}{M_0} \right) N_t({\sf r}) \mid X_0 = x \right] \vert_{{\sf r} = 0}.\]

These second elasticities will cease to be constant as a function of \(t\), capturing a different intertemporal aspect of valuation. Perturbations at intermediate dates are also possible.

9.5. Intertemporal asset price compensations#

As in the previous section, we work with proportional representations of risk compensations. We extend the limiting characterizations by filling in the intertemporal components through perturbations in the cash flow. Thus the objects of interest are:

\[\log {\mathbb E} \left[\frac {G_t}{G_0} N_1({\sf r}) \Biggl| X_0 = x \right] - \log {\mathbb E} \left[\frac {S_t G_t N_1({\sf r})}{S_0 G_0} \Biggl| X_0 = x \right] + \log {\mathbb E} \left[\frac {S_t}{S_0} \Biggl| X_0 = x \right].\]

where \(G\) is the cash-flow payout process and \(S\) is the cumulative stochastic discount factor process. We compute elasticities by differentiating with respect to \({\sf r}.\) The third term, contributed by the stochastic discount factor, does not depend on \({\sf r}\) and drops out of the computation. The following formula gives the risk compensations by horizon, where each term is a special case of \(\epsilon^m(x,t)\):

\[\epsilon^g(x,t) - \epsilon^{sg}(x,t) . \]

The term, \(\epsilon^g(x,t)\) is the exposure elasticity for the stochastic payoff. Recall that this elasticity can also be viewed as an impulse response for the payout process \(G\) based on altering the conditional mean of the date one shock process. While there may be no direct empirical counterpart to these elasticities, they can be viewed as “building blocks” for intertemporal asset prices, and they can be computed directly for fully specified models of asset valuation.

We will report such elasticities in Chapter 11 when analyzing a canonical asset pricing model with production. We also will explore stochastic counterparts to closely related impulse response functions in Chapter 12.

9.6. Relation to stochastic responses#

Example 9.1 illustrates a close connection between shock elasticities and stochastic responses for linear stochastic models. As [Borovička et al., 2014] establish, this link extends, in part, to models in which the state dynamics are modeled as a Markov diffusion:

\[\begin{split}\begin{align*} dX_t & = \mu(X_t) dt + \sigma(X_t) dW_t \\ dY_t & = \nu(X_t) dt + \varsigma(X_t) dW_t. \end{align*}\end{split}\]

where \(W\) is now a \(k\)-dimensional standard Brownian motion. We denote the filtration (intertemporal family of collections of conditioning information events) \({\mathfrak A} \eqdef \left\{ {\mathfrak A}_t : t\ge 0\right\}\) constructed from the Brownian motion and any pertinent date zero information. Let \(\Lambda\) be the variational process for \(X\) used to construct stochastic responses for \(X\) and \(\Delta\) the corresponding process for the responses of \(Y\), as constructed in the discussion of stochastic responses in Chapter 5. To direct attention to the \(i^{th}\) component of \(dW_0,\) which we denote \(dW_0^i,\) let \(\sigma_i\) denote the \(i^{th}\) column of \(\sigma\) and \(\varsigma_i\) the \(i^{th}\) entry of \(\varsigma\), and impose the initial conditions:

\[\begin{align} \Lambda_0^i & = \sigma_i(X_0) \cr \Delta_0^i & = \varsigma_i(X_0) . \end{align}\]

Then the resulting \({\mathbb E} \left( \Delta^i_t \mid {\mathfrak A}_0 \right) dW_0^i\) process informs us how the \(Y_t\) process responds to \(dW_0\). The process of interest, however, is not \(Y\) but instead \(M =\exp(Y)\). By applying a stochastic version of the chain rule, the date \(t\) stochastic response for \(M\) to \(dW_0^i\) is given by

\[{\mathbb E} \left(M_t \Delta_t^i \mid {\mathfrak A}_0 \right) dW_0^i\]

We now deploy the continuous-time counterpart to (9.2)

(9.8)#\[\epsilon^m(x,t) \eqdef \sum_{i=1}^k \pi_i(x) \frac { {\mathbb E} \left[ \left( \frac {M_t}{M_0} \right) \Delta^i_t \mid X_0 =x \right]} { {\mathbb E} \left[ \left( \frac {M_t}{M_0} \right) \mid X_0 = x \right] }\]

where \(\pi_i(x)\) is the \(i^{th}\) entry of \(\pi\). We include the division by \(M_0\) in the numerator and denominator so that we can exploit the Markov structure when computing conditional expectations. As we see, the shock elasticity is a weighted average of the stochastic responses for \(Y\) where the date \(t\) weights are given by:

\[\frac {\left( \frac {M_t}{M_0} \right) } { {\mathbb E} \left[ \left( \frac {M_t}{M_0} \right) \mid X_0 = x \right]}. \]

In summary, we obtain (9.8) as a continuous-time counterpart to (9.2) by adding two steps, i) we measure the exposures of \(M_t\) to \(dW_0^i\) and ii) we use the fact that the local variance of \(dW_0^i\) is specified to be unity.

Finally, suppose that \(M = {\widetilde L}\) is a multiplicative martingale with the initialization \(M_0 = {\widetilde L}_0 = 1\). This will be the case when we study the impacts of changes in probability. Move the perturbation to date \(t\), where it is captured by \(\pi(X_t) \cdot dW_t\). The \(dW_t^i\) stochastic impact is now

\[\begin{split} {\widetilde L}_t \varsigma_i(X_t) dW_t^i. \end{split}\]

Using analogous logic, the shock elasticities are:

\[{\hat \epsilon}^m(x,t) \eqdef \sum_{i=1}^k \pi_i(x) \, {\mathbb E} \left[ {\widetilde L}_t \varsigma_i(X_t) \mid X_0 = x \right] \]

which will depend on the horizon \(t\) provided that \(\varsigma\) is state-dependent.

9.7. Summary#

A shock elasticity is the derivative, with respect to the size of a date-one perturbation, of the logarithm of an expected multiplicative functional. At a one-period horizon it is \(\pi(x)\cdot\kappa_2(x)\). At long horizons the martingale factorization of Chapter 8 delivers the limit, the eigenfunction ratio washing out under stochastic stability. Differencing elasticities computed for a cash flow, for a stochastic discount factor, and for their product yields an intertemporal risk compensation, with exposure to uncertainty separated from the price of that exposure.

Elasticities and the stochastic responses of Chapter 5 answer different questions about the same system: a response is a derivative with respect to a state, an elasticity a derivative with respect to an exposure to a shock. Chapter 11 and Chapter 12 use both.

9.8. Exercises#

Exercise 9.1 (The one-period elasticity)

The chapter states that at the one-period horizon the shock elasticity reduces to \(\epsilon^m(x,1) = \pi(x)\cdot\kappa_2(x)\). Verify it.

Write the multiplicative functional as

\[\log M_{t+1} - \log M_t = \kappa_1(X_t) + \kappa_2(X_t)\cdot W_{t+1}\]

and the perturbation as \(\log N_1({\sf r}) = {\sf r}\,\pi(X_0)\cdot W_1 - {\frac 1 2}\left|\pi(X_0)\right|^2 {\sf r}^2\).

(a) Confirm that \({\mathbb E}\left[N_1({\sf r})\mid X_0 = x\right] = 1\) for every \({\sf r}\), so the perturbation is admissible.

(b) Compute \({\mathbb E}\left[\left(M_1/M_0\right)N_1({\sf r}) \mid X_0 = x\right]\) using properties of the log-normal distribution. Be careful to collect both the \(\kappa_2\) and the \({\sf r}\pi\) loadings on \(W_1\) before squaring.

(c) Differentiate the logarithm of your answer with respect to \({\sf r}\) and evaluate at \({\sf r} = 0\) to obtain \(\epsilon^m(x,1) = \pi(x)\cdot\kappa_2(x)\).

(d) Explain why the term \(-{\frac 1 2}\left|\pi(X_0)\right|^2{\sf r}^2\) makes no contribution to the derivative, and what would go wrong if it were omitted from the definition of \(N_1({\sf r})\) altogether.

Exercise 9.2 (Elasticities for a log-normal vector autoregression)

Revisit Example 9.1. Let

\[X_{t+1} = {\mathbb A}X_t + {\mathbb B}W_{t+1}, \qquad Y_{t+1} - Y_t = \nu + {\mathbb D}X_t + {\mathbb F}W_{t+1},\]

with \({\mathbb A}\) stable and \(\{W_{t+1}\}\) i.i.d. standard normal, and set \(M = \exp(Y)\).

(a) Compute \({\mathbb E}\left(Y_t - Y_0 \mid {\mathfrak A}_1\right)\) and \({\mathbb E}\left(Y_t - Y_0\mid{\mathfrak A}_0\right)\) and take the difference to obtain the date-one forecast error. Check the upper limits of your summations by evaluating at \(t = 1\).

(b) Deduce the elasticity \(\epsilon^m(x,t)\) and verify that at \(t = 1\) it reduces to \(\pi\cdot{\mathbb F}\), consistent with Exercise 9.1.

(c) Evaluate \(\lim_{t\rightarrow\infty}\epsilon^m(x,t)\). Identify the resulting vector as an object from Chapter 4 and say what it measures.

(d) Why does \(\epsilon^m(x,t)\) not depend on the state \(x\) here? Which feature of the specification accounts for this, and what would have to change for state dependence to appear?

Exercise 9.3 (Elasticities of a multiplicative martingale)

Suppose \(M = {\widetilde L}\) is a multiplicative martingale with \({\widetilde L}_0 = 1\), as in the change-of-measure applications of Chapter 8.

(a) Using the Law of Iterated Expectations, show that \(\epsilon^m(x,t)\) defined in (9.1) is the same for every \(t \ge 1\).

(b) Give the economic reading of (a): why should perturbing the date one shock have an effect on a martingale that does not decay, and yet does not grow either?

(c) Now consider the second elasticity (9.7), in which the perturbation is moved to date \(t\) rather than date one. Show that it need not be constant in \(t\), and identify precisely what it depends on.

(d) Under what initialization of the \({\widetilde \cdot}\) process would \(\varepsilon^m(x,t)\) also be constant in \(t\)? Relate your answer to the stochastic stability notion of Chapter 8.

Exercise 9.4 (Risk prices in the log-normal case)

The chapter defines the intertemporal risk compensation as \(\epsilon^g(x,t) - \epsilon^{sg}(x,t)\), where \(G\) is a cash-flow process and \(S\) a cumulative stochastic discount factor. Evaluate it when both are log-normal functionals of the same vector autoregression.

Let \(X_{t+1} = {\mathbb A}X_t + {\mathbb B}W_{t+1}\) and

\[\log G_{t+1} - \log G_t = \nu_g + {\mathbb D}_g X_t + {\mathbb F}_g W_{t+1}, \qquad \log S_{t+1} - \log S_t = \nu_s + {\mathbb D}_s X_t + {\mathbb F}_s W_{t+1} .\]

(a) Write down the additive functional governing \(SG\) and identify its \({\mathbb D}\) and \({\mathbb F}\) coefficients.

(b) Using Exercise 9.2, show that

\[\epsilon^g(x,t) - \epsilon^{sg}(x,t) = -\,\epsilon^s(x,t)\]

for every \(t\), where \(\epsilon^s\) is the exposure elasticity of the stochastic discount factor alone.

(c) Interpret the sign. Why is the price of exposure to a shock the negative of the discount factor’s own exposure to it, and how does this compare with the familiar statement that risk premia are minus a covariance with the stochastic discount factor?

(d) Result (b) says the risk-price elasticity does not depend on the cash flow at all. Explain why this is special to the log-normal case, and identify which term in the general formula (9.4) breaks it when the model is not log-normal.

9.9. Answers#