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Long Term Risk: An Operator Approach

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  • Lars Peter Hansen
  • Jose Scheinkman

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. This family forms a semigroup whose members are indexed by the elapsed time between payoff and valuation dates. 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 establish 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 tradeoff.

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Bibliographic Info

Paper provided by National Bureau of Economic Research, Inc in its series NBER Working Papers with number 12650.

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Date of creation: Oct 2006
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Publication status: published as Lars Peter Hansen & José A. Scheinkman, 2009. "Long-Term Risk: An Operator Approach," Econometrica, Econometric Society, vol. 77(1), pages 177-234, 01.
Handle: RePEc:nbr:nberwo:12650

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  1. Jessica Wachter & Martin Lettau, 2005. "Why is Long-Horizon Equity Less Risky? A Duration-Based Explanation of the Value Premium," 2005 Meeting Papers, Society for Economic Dynamics 302, Society for Economic Dynamics.
  2. David M Kreps & Evan L Porteus, 1978. "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Levine's Working Paper Archive 625018000000000009, David K. Levine.
  3. Bansal, Ravi & Lehmann, Bruce N., 1997. "Growth-Optimal Portfolio Restrictions On Asset Pricing Models," Macroeconomic Dynamics, Cambridge University Press, Cambridge University Press, vol. 1(02), pages 333-354, June.
  4. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, Econometric Society, vol. 68(6), pages 1343-1376, November.
  5. David K. Backus & Stanley E. Zin, 1994. "Reverse Engineering the Yield Curve," NBER Working Papers 4676, National Bureau of Economic Research, Inc.
  6. Lars Peter Hansen & Jose Alexandre Scheinkman, 1993. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," NBER Technical Working Papers 0141, National Bureau of Economic Research, Inc.
  7. 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, MIT Press, vol. 1(1), pages 68-123, 03.
  8. Xiaohong Chen & Lars Peter Hansen & Jos´e A. Scheinkman, 2005. "Principal Components and the Long Run," Levine's Bibliography 122247000000000997, UCLA Department of Economics.
  9. Breeden, Douglas T., 1979. "An intertemporal asset pricing model with stochastic consumption and investment opportunities," Journal of Financial Economics, Elsevier, Elsevier, vol. 7(3), pages 265-296, September.
  10. Nina Boyarchenko & Sergei Levendorski&icaron;, 2007. "The Eigenfunction Expansion Method In Multi-Factor Quadratic Term Structure Models," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 17(4), pages 503-539.
  11. Schroder, Mark & Skiadas, Costis, 1999. "Optimal Consumption and Portfolio Selection with Stochastic Differential Utility," Journal of Economic Theory, Elsevier, Elsevier, vol. 89(1), pages 68-126, November.
  12. repec:cup:macdyn:v:1:y:1997:i:2:p:333-54 is not listed on IDEAS
  13. Fernando Alvarez & Urban J. Jermann, 2005. "Using Asset Prices to Measure the Persistence of the Marginal Utility of Wealth," Econometrica, Econometric Society, Econometric Society, vol. 73(6), pages 1977-2016, November.
  14. Ravi Bansal & Amir Yaron, 2000. "Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles," NBER Working Papers 8059, National Bureau of Economic Research, Inc.
  15. Lars Peter Hansen, 2008. "Modeling the Long Run: Valuation in Dynamic Stochastic Economies," NBER Working Papers 14243, National Bureau of Economic Research, Inc.
  16. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, Econometric Society, vol. 60(2), pages 353-94, March.
  17. Stutzer, Michael, 2003. "Portfolio choice with endogenous utility: a large deviations approach," Journal of Econometrics, Elsevier, Elsevier, vol. 116(1-2), pages 365-386.
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