A Two-State Capital Asset Pricing Model with Unobservable States

We derive theoretical discrete time asset pricing restrictions on the within state conditional mean equations for the market portfolio and for individual assets under the assumptions: (1) the conditional CAPM holds; (2) asset returns are driven by an underlying unobserved two-state discrete Markov process. We show that the market risk-premiums in the two states can be decomposed into a standard CAPM volatility-level premium plus an additional volatility-uncertainty premium. The latter premium is increasing in the market price of risk, the uncertainty about the next period's state and the difference in volatility between the two states. In an empirical application the model is estimated for the U.S. stock market 1836-2003. We apply a discrete mixture of two Normal Inverse Gaussian (NIG) distributions to represent the return characteristics in the unobservable states. Our results show that the high-risk regime has a volatility of 36.28 % on an annual basis while the low-risk regime has just 14.42%, and the latter is much more frequent. Stock returns display characteristics that support our specification of within state NIG distributions as an alternative to Normal distributions. The risk premiums for the two regimes are 2.79% and 17.86% on an annual basis, but the volatility-uncertainty premium for the two states are shown to give an unimportant contribution to the estimated risk premium. The most striking result, from a practical point of view, is that the average sample risk premium of 4% belongs to the highest quintiles of the estimated conditional risk premiums.