Comments from Jaroslav Borovička, Rene Carmona, Vasco Carvalho, Junghoon Lee, Angelo Melino, Eric Renault, Chris Rogers, Mike Stutzer, Grace Tsiang, and Yong Wang were very helpful in preparing this paper. We also benefited from valuable feedback from the co-editor, Larry Samuelson, and the referees of this paper. This material is based on work supported by the National Science Foundation under award numbers SES-05-19372, SES-03-50770, and SES-07-18407. Portions of this work were done while José Scheinkman was a Fellow of the John Simon Guggenheim Memorial Foundation.
Long-Term Risk: An Operator Approach
Article first published online: 15 DEC 2008
© 2009 The Econometric Society
Volume 77, Issue 1, pages 177–234, January 2009
How to Cite
Hansen, L. P. and Scheinkman, J. A. (2009), Long-Term Risk: An Operator Approach. Econometrica, 77: 177–234. doi: 10.3982/ECTA6761
- Issue published online: 15 DEC 2008
- Article first published online: 15 DEC 2008
- Manuscript received October, 2006; final revision received June, 2008.
- Risk-return trade-off;
- long run;
- Perron–Frobenius theory;
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.