Working Paper: NBER ID: w12650
Authors: Lars Peter Hansen; Jose Scheinkman
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.
Keywords: long-term risk; Markov environments; risk-return relationship; valuation operators
JEL Codes: G12
Edges that are evidenced by causal inference methods are in orange, and the rest are in light blue.
| Cause | Effect |
|---|---|
| stochastic growth components (O49) | eigenvalues (D46) |
| stochastic growth components (O49) | eigenfunctions (C29) |
| cash flow risks (G32) | valuation (D46) |
| eigenvalues (D46) | expected return (G17) |
| cash flow risk exposure (G32) | asymptotic rates of return (G17) |
| local risk-return tradeoffs (G11) | cumulative returns (G12) |