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Long-term Risk: An Operator Approach

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  • Lars Peter Hansen
  • Jose A Sheinkman

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.

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Bibliographic Info

Paper provided by UCLA Department of Economics in its series Levine's Bibliography with number 122247000000001669.

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Date of creation: 15 Nov 2007
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Handle: RePEc:cla:levrem:122247000000001669

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  1. 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.
  2. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, Econometric Society, vol. 60(2), pages 353-94, March.
  3. Xiaohong Chen & Lars Peter Hansen & Jos´e A. Scheinkman, 2005. "Principal Components and the Long Run," Levine's Bibliography 122247000000000997, UCLA Department of Economics.
  4. Darrell Duffie & Jun Pan & Kenneth Singleton, 1999. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," NBER Working Papers 7105, National Bureau of Economic Research, Inc.
  5. 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.
  6. Hansen, Lars Peter & Scheinkman, Jose Alexandre, 1995. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," Econometrica, Econometric Society, Econometric Society, vol. 63(4), pages 767-804, July.
  7. Lars Peter Hansen, 2008. "Modeling the Long Run: Valuation in Dynamic Stochastic Economies," NBER Working Papers 14243, National Bureau of Economic Research, Inc.
  8. David K. Backus & Stanley E. Zin, 1994. "Reverse Engineering the Yield Curve," NBER Working Papers 4676, National Bureau of Economic Research, Inc.
  9. David M Kreps & Evan L Porteus, 1978. "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Levine's Working Paper Archive 625018000000000009, David K. Levine.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. repec:cup:macdyn:v:1:y:1997:i:2:p:333-54 is not listed on IDEAS
  15. 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.
  16. 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.
  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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