  This paper investigates how two important sources of risk -- market tail risk
and extreme market volatility risk -- are priced into the cross-section of
asset returns across various investment horizons. To identify such risks, we
propose a quantile spectral beta representation of risk based on the
decomposition of covariance between indicator functions that capture
fluctuations over various frequencies. We study the asymptotic behavior of the
proposed estimators of such risk. Empirically, we find that tail risk is a
short-term phenomenon, whereas extreme volatility risk is priced by investors
in the long term when pricing a cross-section of individual stocks. In
addition, we study popular industry, size and value, profit, investment or
book-to-market portfolios, as well as portfolios constructed from various asset
classes, portfolios sorted on cash flow duration and other strategies. These
results reveal that tail-dependent and horizon-specific risks are priced
heterogeneously across datasets and are important sources of risk for
investors.
