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Long Term Risk

José SCHEINKMAN (Princeton University)

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 semigroup of valuation operators. 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 tradeoff.

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