IDEAS home Printed from https://ideas.repec.org/p/cla/levrem/122247000000001669.html
   My bibliography  Save this paper

Long-term Risk: An Operator Approach

Author

Listed:
  • 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. 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.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Lars Peter Hansen & Jose A Sheinkman, 2007. "Long-term Risk: An Operator Approach," Levine's Bibliography 122247000000001669, UCLA Department of Economics.
  • Handle: RePEc:cla:levrem:122247000000001669
    as

    Download full text from publisher

    File URL: http://www.princeton.edu/%7Ejoses/wp/Long_term.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    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.
    2. repec:cup:macdyn:v:1:y:1997:i:2:p:333-54 is not listed on IDEAS
    3. Michael Stutzer, 2011. "Portfolio choice with endogenous utility: a large deviations approach," World Scientific Book Chapters, in: Leonard C MacLean & Edward O Thorp & William T Ziemba (ed.), THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 43, pages 619-640, World Scientific Publishing Co. Pte. Ltd..
    4. Martin Lettau & Jessica A. Wachter, 2007. "Why Is Long‐Horizon Equity Less Risky? A Duration‐Based Explanation of the Value Premium," Journal of Finance, American Finance Association, vol. 62(1), pages 55-92, February.
    5. 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.
    6. 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.
    7. Larry G. Epstein & Stanley E. Zin, 2013. "Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 12, pages 207-239, World Scientific Publishing Co. Pte. Ltd..
    8. Lars Peter Hansen & Thomas J Sargent, 2014. "A Quartet of Semigroups for Model Specification, Robustness, Prices of Risk, and Model Detection," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 4, pages 83-143, World Scientific Publishing Co. Pte. Ltd..
    9. Xiaohong Chen & Lars Peter Hansen & Jos´e A. Scheinkman, 2005. "Principal Components and the Long Run," Levine's Bibliography 122247000000000997, UCLA Department of Economics.
    10. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-394, March.
    11. 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, August.
    12. Nina Boyarchenko & Sergei Levendorskiǐ, 2007. "The Eigenfunction Expansion Method In Multi‐Factor Quadratic Term Structure Models," Mathematical Finance, Wiley Blackwell, vol. 17(4), pages 503-539, October.
    13. 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.
    14. 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.
    15. David K. Backus & Stanley E. Zin, 1994. "Reverse Engineering the Yield Curve," Working Papers 94-09, New York University, Leonard N. Stern School of Business, Department of Economics.
    16. Lars Peter Hansen & John C. Heaton & Nan Li, 2008. "Consumption Strikes Back? Measuring Long-Run Risk," Journal of Political Economy, University of Chicago Press, vol. 116(2), pages 260-302, April.
    17. 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.
    18. Bansal, Ravi & Lehmann, Bruce N., 1997. "Growth-Optimal Portfolio Restrictions On Asset Pricing Models," Macroeconomic Dynamics, Cambridge University Press, vol. 1(2), pages 333-354, June.
    19. Lars Peter Hansen, 2008. "Modeling the Long Run: Valuation in Dynamic Stochastic Economies," NBER Working Papers 14243, National Bureau of Economic Research, Inc.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lars Peter Hansen, 2008. "Modeling the Long Run: Valuation in Dynamic Stochastic Economies," NBER Working Papers 14243, National Bureau of Economic Research, Inc.
    2. Jaroslav Borovička & Lars Peter Hansen & José A. Scheinkman, 2016. "Misspecified Recovery," Journal of Finance, American Finance Association, vol. 71(6), pages 2493-2544, December.
    3. repec:pri:metric:wp051_2013_hansen_scheinkman_stochastic-compounding-and-uncertain-valuati is not listed on IDEAS
    4. Borovička, Jaroslav & Hansen, Lars Peter, 2014. "Examining macroeconomic models through the lens of asset pricing," Journal of Econometrics, Elsevier, vol. 183(1), pages 67-90.
    5. Jaroslav Borovička & Mark Hendricks & José A. Scheinkman, 2011. "Risk-Price Dynamics," Journal of Financial Econometrics, Oxford University Press, vol. 9(1), pages 3-65, Winter.
      • Jaroslav Borovička & Lars Peter Hansen & Mark Hendricks & José A. Scheinkman, 2009. "Risk Price Dynamics," NBER Working Papers 15506, National Bureau of Economic Research, Inc.
      • Jaroslav Borovicka & Lars Peter Hansen & Mark Hendricks & Jose A. Scheinkman, 2009. "Risk Price Dynamics," Working Papers 1393, Princeton University, Department of Economics, Econometric Research Program..
    6. Lars Peter Hansen & John C. Heaton & Nan Li, 2008. "Consumption Strikes Back? Measuring Long-Run Risk," Journal of Political Economy, University of Chicago Press, vol. 116(2), pages 260-302, April.
    7. Jaroslav Borovička & Mark Hendricks & José A. Scheinkman, 2011. "Risk-Price Dynamics," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 9(1), pages 3-65, Winter.
      • Jaroslav Borovička & Lars Peter Hansen & Mark Hendricks & José A. Scheinkman, 2009. "Risk Price Dynamics," NBER Working Papers 15506, National Bureau of Economic Research, Inc.
      • Lars Peter Hansen & Jaroslav BoroviÄ ka & Mark Hendricks & José A. Scheinkman, 2010. "Risk Price Dynamics," Working Papers 2010-004, Becker Friedman Institute for Research In Economics.
      • Jaroslav Borovicka & Lars Peter Hansen & Mark Hendricks & Jose A. Scheinkman, 2009. "Risk Price Dynamics," Working Papers 1393, Princeton University, Department of Economics, Econometric Research Program..
    8. Hansen, Lars Peter, 2013. "Risk Pricing over Alternative Investment Horizons," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, volume 2, chapter 0, pages 1571-1611, Elsevier.
    9. Hansen, Lars Peter, 2013. "Uncertainty Outside and Inside Economic Models," Nobel Prize in Economics documents 2013-7, Nobel Prize Committee.
    10. repec:pri:metric:wp033_2012_hansen_borovicka_hendricks_scheinkman_risk%20price%20dynamics. is not listed on IDEAS
    11. David Backus & Mikhail Chernov & Stanley Zin, 2014. "Sources of Entropy in Representative Agent Models," Journal of Finance, American Finance Association, vol. 69(1), pages 51-99, February.
    12. Backus, David & Boyarchenko, Nina & Chernov, Mikhail, 2018. "Term structures of asset prices and returns," Journal of Financial Economics, Elsevier, vol. 129(1), pages 1-23.
    13. Jaroslav Borovička & Lars P. Hansen & Jose A. Scheinkman, 2014. "Shock Elasticities and Impulse Responses," NBER Working Papers 20104, National Bureau of Economic Research, Inc.
    14. Matthijs Breugem & Stefano Colonello & Roberto Marfè & Francesca Zucchi, 2020. "Dynamic Equity Slope," Working Papers 2020:21, Department of Economics, University of Venice "Ca' Foscari".
    15. Neuhierl, Andreas & Varneskov, Rasmus T., 2021. "Frequency dependent risk," Journal of Financial Economics, Elsevier, vol. 140(2), pages 644-675.
    16. Hengjie Ai & Ravi Bansal, 2016. "Risk Preferences and The Macro Announcement Premium," NBER Working Papers 22527, National Bureau of Economic Research, Inc.
    17. Borovička, Jaroslav & Hansen, Lars Peter, 2014. "Examining macroeconomic models through the lens of asset pricing," Journal of Econometrics, Elsevier, vol. 183(1), pages 67-90.
    18. Tsai, Jerry & Wachter, Jessica A., 2018. "Pricing long-lived securities in dynamic endowment economies," Journal of Economic Theory, Elsevier, vol. 177(C), pages 848-878.
    19. Anisha Ghosh & Christian Julliard & Alex P. Taylor, 2017. "What Is the Consumption-CAPM Missing? An Information-Theoretic Framework for the Analysis of Asset Pricing Models," The Review of Financial Studies, Society for Financial Studies, vol. 30(2), pages 442-504.
    20. Roussanov, Nikolai, 2014. "Composition of wealth, conditioning information, and the cross-section of stock returns," Journal of Financial Economics, Elsevier, vol. 111(2), pages 352-380.
    21. Yu Chen & Thomas Cosimano & Alex Himonas & Peter Kelly, 2014. "An Analytic Approach for Stochastic Differential Utility for Endowment and Production Economies," Computational Economics, Springer;Society for Computational Economics, vol. 44(4), pages 397-443, December.
    22. Campani, Carlos Heitor & Garcia, René, 2019. "Approximate analytical solutions for consumption/investment problems under recursive utility and finite horizon," The North American Journal of Economics and Finance, Elsevier, vol. 48(C), pages 364-384.

    More about this item

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cla:levrem:122247000000001669. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: David K. Levine (email available below). General contact details of provider: http://www.dklevine.com/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.