TY  - JOUR
AU  - Hansen,Lars Peter
AU  - Scheinkman,Jose
TI  - Long Term Risk: An Operator Approach
JF  - National Bureau of Economic Research Working Paper Series
VL  - No. 12650
PY  - 2006
Y2  - October 2006
UR  - http://www.nber.org/papers/w12650
L1  - http://www.nber.org/papers/w12650.pdf
N1  - Author contact info:
Lars P. Hansen
Department of Economics
The University of Chicago
1126 East 59th Street
Chicago, IL 60637
Tel: 773/702-8170
Fax: 773/702-8490
E-Mail: lhansen@uchicago.edu

Jose A. Scheinkman
Department of Economics
Princeton University
Princeton, NJ  08544-1021
Tel: 609/258-4020
Fax: 609/258-0771
E-Mail: joses@princeton.edu
AB  - 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.
ER  -