Long-Term Risk: An Operator Approach
AbstractWe 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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Bibliographic InfoArticle provided by Econometric Society in its journal Econometrica.
Volume (Year): 77 (2009)
Issue (Month): 1 (01)
Other versions of this item:
- Hansen, Lars Peter & Scheinkman, José A., 2008. "Long Term Risk : An Operator Approach," Economics Papers from University Paris Dauphine 123456789/2282, Paris Dauphine University.
- Lars Peter Hansen & Jose A Sheinkman, 2007. "Long-term Risk: An Operator Approach," Levine's Bibliography 122247000000001669, UCLA Department of Economics.
- Lars Peter Hansen & Jose Scheinkman, 2006. "Long Term Risk: An Operator Approach," NBER Working Papers 12650, National Bureau of Economic Research, Inc.
- G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill
- D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
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