July 2012 | Article

Recursive Utility in a Markov Environment with Stochastic Growth

Lars Peter Hansen, Jose A. Sheinkman

Recursive utility models that feature investor concerns about the intertemporal composition of risk are used extensively in applied research in macroeconomics and asset pricing. These models represent preferences as the solution to a nonlinear forward-looking difference equation with a terminal condition. In this paper we study infinite-horizon specifications of this difference equation in the context of a Markov environment. We establish a connection between the solution to this equation and to an arguably simpler Perron–Frobenius eigenvalue equation of the type that occurs in the study of large deviations for Markov processes. By exploiting this connection, we establish existence and uniqueness results. Moreover, we explore a substantive link between large deviation bounds for tail events for stochastic consumption growth and preferences induced by recursive utility.

Journal: Proceedings of the National Academy of Sciences|Volume: 109|Issue Number: 30|Pages: 11967-11972|Tags: Financial Market Linkages to the Macroeconomy, Uncertainty and Valuation|Export BibTeX >
  title={Recursive Utility in a Markov Environment With Stochastic Growth},
  author={Hansen, Lars Peter and Scheinkman, Jos{'e} A},
  journal={Proceedings of the National Academy of Sciences},
  publisher={National Acad Sciences}
May 2012 | Article

Dynamic Valuation Decomposition Within Stochastic Economies: Fisher–Schultz Lecture

Lars Peter Hansen

I explore the equilibrium value implications of economic models that incorporate responses to a stochastic environment with growth. I propose dynamic valuation decompositions (DVD’s) designed to distinguish components of an underlying economic model that influence values over long investment horizons from components that impact only the short run. A DVD represents the values of stochastically growing claims to consumption payoffs or cash flows using a stochastic discount process that both discounts the future and adjusts for risk. It is enabled by constructing operators indexed by the elapsed time between the trading date and the date of the future realization of the payoff. Thus formulated, methods from applied mathematics permit me to characterize valuation behavior and the term structure of risk prices in a revealing manner. I apply this approach to investigate how investor beliefs and the associated uncertainty are reflected in current-period values and risk-price elasticities.

Journal: Econometrica|Volume: 80|Issue Number: 3|Pages: 911-967|Tags: Uncertainty and Valuation|Export BibTeX >

  title={Dynamic Valuation Decomposition Within Stochastic Economies},
  author={Hansen, Lars Peter},
  publisher={Wiley Online Library}
April 2012 | Article

Small Noise Methods for Risk-Sensitive/Robust Economies

Evan W. Anderson, Lars Peter Hansen, Thomas J. Sargent

We provide small noise expansions for the value function and decision rule for the recursive risk-sensitive preferences specified by Hansen and Sargent (1995), Hansen et al. (1999), and Tallarini (2000). We use the expansions (1) to provide a fast method for approximating solutions of dynamic stochastic problems and (2) to quantify the effects on decisions of uncertainty and concerns about robustness to misspecification.

Journal: Journal of Economic Dynamics & Control|Volume: 36|Issue Number: 4|Pages: 468-500|Tags: Risk, Robustness and Ambiguity|Export BibTeX >
  title={Small Noise Methods for Risk-Sensitive/Robust economies},
  author={Anderson, Evan W. and Hansen, Lars Peter and Sargent, Thomas J.},
  journal={Journal of Economic Dynamics and Control},
January 2012 | Article

Pricing Growth-Rate Risk

Lars Peter Hansen, José A. Scheinkman

We characterize the compensation demanded by investors in equilibrium for incremental exposure to growth-rate risk. Given an underlying Markov diffusion that governs the state variables in the economy, the economic model implies a stochastic discount factor process S. We also consider a reference growth process G that may represent the growth in the payoff of a single asset or of the macroeconomy. Both S and G are modeled conveniently as multiplicative functionals of a multidimensional Brownian motion. We consider the pricing implications of parametrized family of growth processes G , with G0 = G, as is made small. This parametrization defines a direction of growth-rate risk exposure that is priced using the stochastic discount factor S. By changing the investment horizon, we trace a term structure of risk prices that shows how the valuation of risky cash flows depends on the investment horizon. Using methods of Hansen and Scheinkman (Econometrica 77:177–234, 2009), we characterize the limiting behavior of the risk prices as the investment horizon is made arbitrarily long.

Journal: Finance and Stochastics|Volume: 16|Issue Number: 1|Pages: 1-15|Tags: Uncertainty and Valuation|Export BibTeX >
  title={Pricing Growth-Rate Risk},
  author={Hansen, Lars Peter and Scheinkman, Jos{'e} A},
  journal={Finance and Stochastics},
May 2011 | Article

Robustness and Ambiguity in Continuous Time

Lars Peter Hansen, Thomas J. Sargent

We use statistical detection theory in a continuous-time environment to provide a new perspective on calibrating a concern about robustness or an aversion to ambiguity. A decision maker repeatedly confronts uncertainty about state transition dynamics and a prior distribution over unobserved states or parameters. Two continuous-time formulations are counterparts of two discrete-time recursive specifications of Hansen and Sargent (2007) [16]. One formulation shares features of the smooth ambiguity model of Klibanoff et al. (2005) and (2009) [24] and [25]. Here our statistical detection calculations guide how to adjust contributions to entropy coming from hidden states as we take a continuous-time limit.

Journal: Journal of Economic Theory|Volume: 146|Issue Number: 3|Pages: 1195-1223|Tags: Risk, Robustness and Ambiguity|Export BibTeX >
  title={Robustness and Abiguity in Continuous Time},
  author={Hansen, Lars Peter and Sargent, Thomas J.},
  journal={Journal of Economic Theory},
January 2011 | Article

Risk Price Dynamics

Lars Peter Hansen, Jaroslav Borovička, Mark Hendricks, José A. Scheinkman

We present a novel approach to depicting asset-pricing dynamics by characterizing shock exposures and prices for alternative investment horizons. We quantify the shock exposures in terms of elasticities that measure the impact of a current shock on future cash flow growth. The elasticities are designed to accommodate nonlinearities in the stochastic evolution modeled as a Markov process. Stochastic growth in the underlying macroeconomy and stochastic discounting in the representation of asset values are central ingredients in our investigation. We provide elasticity calculations in a series of examples featuring consumption externalities, recursive utility, and jump risk.

This paper was originally presented as the Journal of Financial Econometrics Lecture at the June 2009 SoFiE conference.

Journal: Journal of Financial Econometrics|Volume: 9|Issue Number: 1|Pages: 3-65|Tags: Uncertainty and Valuation|Export BibTeX >
  title={Risk-Price Dynamics},
  author={Borovi{v{c}}ka, Jaroslav and Hansen, Lars Peter and Hendricks, Mark and Scheinkman, Jos{'e} A},
  journal={Journal of Financial Econometrics},
  publisher={Oxford Univ Press}
December 2010 | Chapter

Wanting Robustness in Macroeconomics

Lars Peter Hansen, Thomas J. Sargent

Robust control theory is a tool for assessing decision rules when a decision maker distrusts either the specification of transition laws or the distribution of hidden state variables or both. Specification doubts inspire the decision maker to want a decision rule to work well for a ? of models surrounding his approximating stochastic model. We relate robust control theory to the so-called multiplier and constraint preferences that have been used to express ambiguity aversion. Detection error probabilities can be used to discipline empirically plausible amounts of robustness. We describe applications to asset pricing uncertainty premia and design of robust macroeconomic policies.

Pages: 1097-1157|Title of book: Handbook of Monetary Economics|Editor(s): Benjamin Friedman, Michael Woodford|Place of Publication: Burlington, MA|Publisher: Elsevier Science|Tags: Risk, Robustness and Ambiguity|Export BibTeX >
  title={Wanting Robustness in Macroeconomics},
  author={Hansen, Lars Peter and Sargent, Thomas J. and others},
  journal={Manuscript, Department of Economics, Stanford University. Website: www. stanford. edu/sargent},
October 2010 | Article

Robust Hidden Markov LQG Problems

Lars Peter Hansen, Ricardo Mayer, Thomas Sargent

For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes’ law. Duality of risk sensitivity to the multiplier version of min–max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I–III, the minimizing players’ worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.

Journal: Journal of Economic Dynamics & Control|Volume: 34|Issue Number: 10|Pages: 1951-1966|Tags: Risk, Robustness and Ambiguity|Export BibTeX >
  title={Robust Hidden Markov LQG Problems},
  author={Hansen, Lars Peter and Mayer, Ricardo and Sargent, Thomas},
  journal={Journal of Economic Dynamics and Control},
October 2010 | Working Paper

Modeling and Measuring Systemic Risk

Markus Brunnermeier, Lars Peter Hansen, Anil Kachyap, Arvind Krishnamurthy, Andrew W. Lo

An important challenge worthy of NSF support is to quantify systemic financial risk. There are at least three major components to this challenge: modeling, measurement, and data accessibility. Progress on this challenge will require extending existing research in many directions and will require collaboration between economists, statisticians, decision theorists, sociologists, psychologists, and neuroscientists.

Tags: Financial Market Linkages to the Macroeconomy|Export BibTeX >
  title={Modeling and Measuring Systemic Risk},
  author={Brunnermeier, Markus K. and Hansen, Lars Peter and Kashyap, Anil K. and Krishnamurthy, Arvind and Lo, Andrew W},
July 2010 | Article

Fragile Beliefs and the Price of Model Uncertainty

Lars Peter Hansen, Thomas J. Sargent

A representative consumer uses Bayes’ law to learn about parameters of several models and to construct probabilities with which to perform ongoing model averaging. The arrival of signals induces the consumer to alter his posterior distribution over models and parameters. The consumer’s specification doubts induce him to slant probabilities pessimistically. The pessimistic probabilities tilt toward a model that puts long-run risks into consumption growth. That contributes a countercyclical history-dependent component to prices of risk.

Journal: Quantitative Economics|Volume: 1|Issue Number: 1|Pages: 129-162|Tags: Financial Market Linkages to the Macroeconomy, Risk, Robustness and Ambiguity|Export BibTeX >
  title={Fragile Beliefs and the Price of Uncertainty},
  author={Hansen, Lars Peter and Sargent, Thomas J.},
  journal={Quantitative Economics},
  publisher={Wiley Online Library}