Frequency Dependent Risk

This paper provides a new nonparametric framework for studying the dynamics of the state vector and its associated risk prices. Specifically, in a general setting where the stochastic discount factor (SDF) decomposes into permanent and transitory components, we analyze their contribution to the unconditional asset return premium using frequency domain techniques. We show analytically that the co-spectrum between returns and the SDF only displays frequency dependencies through its transitory component, that is, through the state vector. Moreover, we demonstrate that state vector dynamics and its risk prices can be uncovered by studying (transformations of) the covariance between (portfolios of) asset returns. We introduce two new frequency risk measures and apply our framework to study its pricing in the full cross-section of US stocks, utilizing the market, value, size and momentum factors as baseline portfolios to construct the measures. Our analysis uncovers the existence of, at least, two significantly priced low-frequency risk factors, one of which commands a large positive risk premium of 6% per year. Moreover, we document, at least, one high-frequency component in the state vector that is significantly priced. Importantly, we show that these frequency dependent risk factors are unspanned by a battery of appraised risk factors and characteristics. Our analysis demonstrates that multiple state vector components with varying persistence and risk prices are needed to be consistent with the cross-section. Throughout, we contrast our findings with the implications of the long-run risk model, the dynamic disaster model as well as a regime-switching CCAPM, providing new analytical results for such models.