Stochastic Compounding and Uncertain Valuation∗

Exploring long-term implications of valuation leads us to recover and use a distorted probability measure that reflects the long-term implications for risk pricing. This measure is typically distinct from the physical and the risk neutral measures that are well known in mathematical finance. We apply a generalized version of Perron-Frobenius theory to construct this probability measure and present several applications. We employ Perron-Frobenius methods to i) explore the observational implications of risk adjustments and investor beliefs as reflected in asset market data; ii) catalog alternative forms of misspecification of parametric valuation models; and iii) characterize how long-term components of growth-rate risk impact investor preferences implied by Kreps-Porteus style utility recursions.