Risk Pricing over Alternative Investment Horizons
I explore methods that characterize model-based valuation of stochastically growing cash flows. Following previous research, I use stochastic discount factors as a convenient device to depict asset values. I extend that literature by focusing on the impact of compounding these discount factors over alternative investment horizons. In modeling cash flows, I also incorporate stochastic growth factors. I explore dynamic value decomposition (DVD) methods that capture concurrent compounding of a stochastic growth and discount factors in determining risk-adjusted values. These methods are supported by factorizations that extract martingale components of stochastic growth and discount factors. These components reveal which ingredients of a model have long-term implications for valuation. The resulting martingales imply convenient changes in measure that are distinct from those used in mathematical finance, and they provide the foundations for analyzing model-based implications for the term structure of risk prices. As an illustration of the methods, I re-examine some recent preference based models. I also use the martingale extraction to revisit the value implications of some benchmark models with market restrictions and heterogenous consumers.
@article{hansen:2012risk,
title={Risk Pricing Over Alternative Investment Horizons},
author={Hansen, Lars Peter},
year={2012}
}
✕ Challenges in Identifying and Measuring Systemic Risk
Sparked by the recent “great recession” and the role of financial markets, considerable interest exists among researchers within both the academic community and the public sector in modeling and measuring systemic risk. In this essay I draw on experiences with other measurement agendas to place in perspective the challenge of quantifying systemic risk, or more generally, of providing empirical constructs that can enhance our understanding of linkages between financial markets and the macroeconomy.
@techreport{hansen:2012challenges,
title={Challenges in Identifying and Measuring Systemic Risk},
author={Hansen, Lars Peter},
year={2012},
institution={National Bureau of Economic Research}
}
✕ Proofs for Large Sample Properties of Generalized Method of Moments Estimators
I present proofs for the consistency of generalized method of moments (GMM) estimators presented in Hansen (1982). Some basic approximation results provide the groundwork for the analysis of a class of such estimators. Using these results, I establish the large sample convergence of GMM estimators under alternative restrictions on the estimation problem.
@article{hansen:2012proofs,
title={Proofs for Large Sample Properties of Generalized Method of Moments Estimators},
author={Hansen, Lars Peter},
journal={Journal of Econometrics},
volume={170},
number={2},
pages={325--330},
year={2012},
publisher={Elsevier}
}
✕ Underidentification?
We develop methods for testing that an econometric model is underidentified and for estimating the nature of the failed identification. We adopt a generalized-method-of moments perspective in a possibly non-linear econometric specification. If, after attempting to replicate the structural relation, we find substantial evidence against the overidentifying restrictions of an augmented model, this is evidence against underidentification of the original model. To diagnose how identification might fail, we study the estimation of a one-dimensional curve that gives the parameter configurations that provide the greatest challenge to identification, and we illustrate this calculation in an empirical example.
@article{arellanohansensentana:2012,
title={Underidentification?},
author={Arellano, Manuel and Hansen, Lars Peter and Sentana, Enrique},
journal={Journal of Econometrics},
volume={170},
number={2},
pages={256–280},
year={2012},
publisher={Elsevier}
}
✕Three Types of Ambiguity
For each of three types of ambiguity, we compute a robust Ramsey plan and an associated worst-case probability model. Ex post, ambiguity of type I implies endogenously distorted homogeneous beliefs, while ambiguities of types II and III imply distorted heterogeneous beliefs. Martingales characterize alternative probability specifications and clarify distinctions among the three types of ambiguity. We use recursive formulations of Ramsey problems to impose local predictability of commitment multipliers directly. To reduce the dimension of the state in a recursive formulation, we transform the commitment multiplier to accommodate the heterogeneous beliefs that arise with ambiguity of types II and III. Our formulations facilitate comparisons of the consequences of these alternative types of ambiguity.
@article{hansensargent:2012three,
title={Three Types of Ambiguity},
author={Hansen, Lars Peter and Sargent, Thomas J.},
journal={Journal of Monetary Economics},
volume={59},
number={5},
pages={422--445},
year={2012},
publisher={Elsevier}
}
✕ Recursive Utility in a Markov Environment with Stochastic Growth
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.
@article{hansenscheinkman:2012recursive,
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},
volume={109},
number={30},
pages={11967--11972},
year={2012},
publisher={National Acad Sciences}
}
✕ Dynamic Valuation Decomposition Within Stochastic Economies: Fisher–Schultz Lecture
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.
@article{hansen:2012dynamic,
title={Dynamic Valuation Decomposition Within Stochastic Economies},
author={Hansen, Lars Peter},
journal={Econometrica},
volume={80},
number={3},
pages={911--967},
year={2012},
publisher={Wiley Online Library}
}
✕ Small Noise Methods for Risk-Sensitive/Robust Economies
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.
@article{ahs:2012small,
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},
volume={36},
number={4},
pages={468--500},
year={2012},
publisher={Elsevier}
}
✕ Pricing Growth-Rate Risk
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.
@article{hansenscheinkman:2012pricing,
title={Pricing Growth-Rate Risk},
author={Hansen, Lars Peter and Scheinkman, Jos{'e} A},
journal={Finance and Stochastics},
volume={16},
number={1},
pages={1--15},
year={2012},
publisher={Springer}
}
✕ Robustness and Ambiguity in Continuous Time
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.
@article{hansensargent:2011robustness,
title={Robustness and Abiguity in Continuous Time},
author={Hansen, Lars Peter and Sargent, Thomas J.},
journal={Journal of Economic Theory},
volume={146},
number={3},
pages={1195--1223},
year={2011},
publisher={Elsevier}
}
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