Long Term Risk: An Operator Approach

Lars Peter Hansen and Jose Scheinkman ()

No 12650, NBER Working Papers from National Bureau of Economic Research, Inc

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.

JEL-codes: G12 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-cfn
Date: Written
Note: AP
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Related works:
Working Paper: Long-term Risk: An Operator Approach (2007)
Journal Article: Long-Term Risk: An Operator Approach (2009)
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