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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Vydáno v:Econometrica Ročník 77; číslo 1; s. 177 - 234
Hlavní autoři: Hansen, Lars Peter, Scheinkman, José A.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Oxford, UK Blackwell Publishing Ltd 01.01.2009
Econometric Society
Wiley-Blackwell
Témata:
Applications
Approximation
Biology, psychology, social sciences
Capital returns
Cash flow
Econometrics
Eigenfunctions
Eigenvalues
Exact sciences and technology
Families & family life
Financial risk
Group theory
Group theory and generalizations
Insurance, economics, finance
Investment risk
long run
Markov analysis
Markov processes
Markovian processes
Martingale
Martingales
Mathematics
Non-linear models
Operators
Perron-Frobenius theory
Probability and statistics
Probability theory and stochastic processes
Risk
Risk adjustment
Risk management
Risk-return trade-off
Sciences and techniques of general use
Semigroups
Statistics
Stochastic processes
Studies
Trade-off
Uniqueness
Valuation
Eigenfunction
Eigenvalue
Continuous time
Probability distribution
Long term variation
semigroups
Transients
martingales
Eigenvalue problem
Eigenvalues and eigenfunctions
Stochastic flow
Existence condition
Approximation
long run
Martingale
Semigroup
Perron-Frobenius theory
Statistical method
Sufficient condition
Capture probability
Risk-return trade-off
Environment
Econometrics
Probability measure
Martingale measure
ISSN:0012-9682, 1468-0262
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Popis
Shrnutí: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.
Bibliografie:istex:134E4C308E4A950254128409293C6318F636A49F
ArticleID:ECTA903
ark:/67375/WNG-02GDT1S9-7
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.
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ISSN:0012-9682
1468-0262
DOI:10.3982/ECTA6761