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Long-Term Risk: An Operator Approach

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Author Info
Lars Peter Hansen
José A. Scheinkman

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Abstract

We create an analytical structure that reveals the long-run risk-return relationship for nonlinear continuous-time Markov environments. We do so by studying an eigenvalue problem associated with a positive eigenfunction for a conveniently chosen family of valuation operators. The members of this family are indexed by the elapsed time between payoff and valuation dates, and they are necessarily related via a mathematical structure called a semigroup. We represent the semigroup using a positive process with three components: an exponential term constructed from the eigenvalue, a martingale, and a transient eigenfunction term. The eigenvalue encodes the risk adjustment, the martingale alters the probability measure to capture long-run approximation, and the eigenfunction gives the long-run dependence on the Markov state. We discuss sufficient conditions for the existence and uniqueness of the relevant eigenvalue and eigenfunction. By showing how changes in the stochastic growth components of cash flows induce changes in the corresponding eigenvalues and eigenfunctions, we reveal a long-run risk-return trade-off. Copyright 2009 The Econometric Society.

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File URL: http://hdl.handle.net/10.3982/ECTA6761
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Publisher Info
Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 77 (2009)
Issue (Month): 1 (01)
Pages: 177-234
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Handle: RePEc:ecm:emetrp:v:77:y:2009:i:1:p:177-234

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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Breeden, Douglas T., 1979. "An intertemporal asset pricing model with stochastic consumption and investment opportunities," Journal of Financial Economics, Elsevier, vol. 7(3), pages 265-296, September. [Downloadable!] (restricted)
  2. Xiaohong Chen & Lars Peter Hansen & Jos´e A. Scheinkman, 2005. "Principal Components and the Long Run," Levine's Bibliography 122247000000000997, UCLA Department of Economics. [Downloadable!]
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  3. repec:cup:macdyn:v:1:y:1997:i:2:p:333-54 is not listed on IDEAS
  4. Ravi Bansal & Amir Yaron, 2004. "Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles," Journal of Finance, American Finance Association, vol. 59(4), pages 1481-1509, 08. [Downloadable!] (restricted)
    Other versions:
  5. Kreps, David M & Porteus, Evan L, 1978. "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica, Econometric Society, vol. 46(1), pages 185-200, January. [Downloadable!] (restricted)
  6. Fernando Alvarez & Urban J. Jermann, 2005. "Using Asset Prices to Measure the Persistence of the Marginal Utility of Wealth," Econometrica, Econometric Society, vol. 73(6), pages 1977-2016, November. [Downloadable!] (restricted)
  7. David K. Backus & Stanley E. Zin, 1994. "Reverse Engineering the Yield Curve," NBER Working Papers 4676, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  8. Martin Lettau & Jessica Wachter, 2005. "Why is Long-Horizon Equity Less Risky? A Duration-Based Explanation of the Value Premium," NBER Working Papers 11144, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  9. Hansen, Lars Peter & Scheinkman, Jose Alexandre, 1995. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," Econometrica, Econometric Society, vol. 63(4), pages 767-804, July. [Downloadable!] (restricted)
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  10. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
  11. Schroder, Mark & Skiadas, Costis, 1999. "Optimal Consumption and Portfolio Selection with Stochastic Differential Utility," Journal of Economic Theory, Elsevier, vol. 89(1), pages 68-126, November. [Downloadable!] (restricted)
  12. Bansal, Ravi & Lehmann, Bruce N., 1997. "Growth-Optimal Portfolio Restrictions On Asset Pricing Models," Macroeconomic Dynamics, Cambridge University Press, vol. 1(02), pages 333-354, June. [Downloadable!]
  13. Evan W. Anderson & Lars Peter Hansen & Thomas J. Sargent, 2003. "A Quartet of Semigroups for Model Specification, Robustness, Prices of Risk, and Model Detection," Journal of the European Economic Association, MIT Press, vol. 1(1), pages 68-123, 03. [Downloadable!] (restricted)
  14. Lars Peter Hansen, 2008. "Modeling the Long Run: Valuation in Dynamic Stochastic Economies," NBER Working Papers 14243, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
  15. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-94, March. [Downloadable!] (restricted)
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Bernard Dumas & Alexander Kurshev & Raman Uppal, 2007. "Equilibrium Portfolio Strategies in the Presence of Sentiment Risk and Excess Volatility," NBER Working Papers 13401, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
  2. Pauline Barrieu & Harry Bensusan & Nicole El Karoui & Caroline Hillairet & Stéphane Loisel & Claudia Ravanelli & Yahia Salhi, 2009. "Understanding, Modeling and Managing Longevity Risk: Key Issues and Main Challenges," Working Papers hal-00417800_v1, HAL. [Downloadable!]
  3. Lars Peter Hansen, 2007. "Beliefs, Doubts and Learning: Valuing Economic Risk," NBER Working Papers 12948, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
  4. Dumas, Bernard J & Kurshev, Alexander & Uppal, Raman, 2007. "Equilibrium Portfolio Strategies in the Presence of Sentiment Risk and Excess Volatility," CEPR Discussion Papers 6455, C.E.P.R. Discussion Papers. [Downloadable!] (restricted)
  5. Xavier Gabaix, 2007. "Linearity-Generating Processes: A Modelling Tool Yielding Closed Forms for Asset Prices," NBER Working Papers 13430, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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