Long-Term Risk: An Operator Approach

Long-Term Risk: An Operator Approach

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 fam...

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Bibliographische Detailangaben
Veröffentlicht in:Econometrica. - Wiley. - 77(2009), 1, Seite 177-234
1. Verfasser: Hansen, Lars Peter (VerfasserIn)
Weitere Verfasser: Scheinkman, José A.
Format: Online-Aufsatz
Sprache:English
Veröffentlicht: 2009
Zugriff auf das übergeordnete Werk:Econometrica
Schlagworte:Risk-return trade-off long run semigroups Perron-Frobenius theory martingales Philosophy Mathematics Economics Business
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520 |a 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. 
540 |a Copyright 2009 The Econometric Society 
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