Intro

Goal: Characterize the main conceptual composition of risk and shock elasticities and provide an example of their interpretation in a macro model.

Outline:

• General continuous time framework
  • Stochastic process for state
  • Growth processes and cash flows
  • Perturbations
  • Elasticities
• Simple example
  • Do the BL model. Can talk about choosing α_d = e_i for i = 1, 2, 3.

Mathematical Framework

I will briefly explain the mathematical framework used in this and related papers.

State

State is Markov diffusion: dX_t = μ(X_t)dt + σ(X_t)dW_t (W is standard BM, μ and σ may be non-linear.)

Functionals

Additive functionals: parameterized by state dependent drift β(X) and volatility α(X) coefficients integrated over time (A_t = ∫₀^tβ(X_u)du + ∫₀^tα(X_u) ⋅ dW_u)

Multiplicative functionals are also characterized by (β, α) and are the exponential of an underlying additive functional (M = exp (A))

Growth processes of economic interest, e.g. cash flows and consumption, will be modeled as multiplicative functionals.

Perturbations

We will perturb the multiplicative functionals in a specific way that allow us to capture the marginal value of exposure risk in a particular direction.

Similar in spirit to a risk premium, which measures the average value of risk exposure (not marginal or isolated to a specific direction.)

How to construct Perturbations

NOTE: only talk about this if I have cruised through the above. I want to be at this point after 2-3 minutes.

We construct perturbations of a multiplicative functional M as the product of M and another multiplicative functional with precisely chosen drift and volatility coefficients. The perturbation coefficients are functions of a scalar parameter r. When r = 0 the perturbation is equal to 1 (i.e. perturbed cash flow is just the cash flow).

Often the drift is chosen so the perturbation itself is a martingale, or the perturbed cash flow is a martingale. This simplifies computation.

The volatility coefficient can be varied to isolate the impact of innovations to a particular dimension of the BM. An example of this will be shown later.

Risk elasticities

A risk elasticity for a cash flow M is defined as the derivative of the log expectation of the perturbed cash flow evaluated at r = 0.

The direct representation of the risk elasticity is additive over a particular time horizon t. To consider contributions that are localized in time, we build an integral representation.

This integral representation expresses the risk elasticity from a given initial state x over a time horizon t as the expectation (conditional on x), of the cash flow in time t times the integral of a function.

The function that is integrated over is called a shock elasticity function and captures the instantaneous contribution of the perturbation to the risk elasticity (i.e. shock elasticities are the building blocks for risk elasticities).

Example

I will now highlight how these theoretical building blocks can be used within a macro model.

Model

The state is 2 dimensional: x₁ represents consumption growth, and x₂ represents stochastic volatility.

The Brownian motion will be three dimensional, where dimensions correspond to independent shocks directly to consumption itself, consumption growth x₁, and stochastic volatility x₂

Multiplicative functionals take a specific form:

• The drift is affine in deviations of x₁ and x₂ from their mean values. The intercept and coefficients on deviations distinguish multiplicative functionals (β(x) = β̄₀ + β̄_t ⋅ (x₁ − ι₁) + β̄₂(x_t − ι₂))
• The volatility is the square root of x₂ times a constant vector $(\alpha(x) = \sqrt{x_2} \bar{\alpha}$).

Consumption growth is modeled directly as a multiplicative functional.

Preferences

Agents have either CRRA preferences or Epstein-Zin preferences over streams of consumption.

Part of the solution to the agent’s problem is a stochastic discount factor used to price cash flows in the economy.

The cash flow of interest will be the growth of the consumption process (C_t/C₀).

Elasticities

We consider two types of fundamental elasticities, as well as a third that is a function of the former two. These types are

1. Exposure elasticities, where the multiplicative function is consumption
2. Value elasticities, where the MF is the price of consumption (consumption times SDF)

We will also consider a price elasticity, which is the difference between the exposure elasticity and the value elasticity. The price elasticities are the functions that characterize the marginal value of exposure risk in a particular direction that we mentioned earlier.

We will allow the volatility coefficient of the perturbation to be each of the vectors [100]′, [010]′, or [001]′. This allows us to construct price elasticities in response to shocks to consumption, consumption growth, and stochastic volatility. This is the sense in which price elasticities characterize the exposure risk in a particular direction.

Results

The exposure elasticities for the two sets of preferences are the same. Remember the exposure elasticity is the elasticity for the cash flow, which is the same underlying consumption process in both models.

The Shock exposure elasticities (local increments) to each of the three dimensions of W are reported.

• Shock to consumption has permanent effect on consumption and the shock elasticity is constant over time
• The shock exposure elasticity to a positive shock in the growth rate increases over time as the impact of a higher growth rate compounds
• The volatility shock exposure elasticity declines over time as the volatility process mean reverts and the impact of the initial shock wears off.

Given the form of agent’s SDF, the shock value elasticities are scaled versions of the shock exposure elasticities.

References