**Working Papers, Department of Economics, Lund University**
# No 2004:28:

A Two-State Capital Asset Pricing Model with Unobservable States

*Birger Nilsson *()

* and Björn Hansson *()

**Abstract:** 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.

**Keywords:** asset pricing; state dependent risk premium; discrete mixture distribution; (follow links to similar papers)

**JEL-Codes:** C22; G12; (follow links to similar papers)

23 pages, December 6, 2004

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