Social states of belief and the determinants of the equity risk premium in a rational belief equilibrium

In previous models of rational belief equilibria (RBE), individual states of belief were the foundation for the construction of the endogenous state space where individual states of belief were described with the method of assessment variables. This leads to a lack of "anonymity" where the belief of each individual agent has an impact on equilibrium prices but as a competitor he ignores it. Instead we study a replica economy with a finite number of types but with a large number of agents of each type. The state space for this economy is constructed as the set of products of the exogenous states and the social states of belief which are vectors of "type-states" each of which is a distribution of beliefs among members of a type. Such an economy leads to RBE which do indeed solve the problem of anonymity. We then study via simulations the implications of the model for market volatility and for the determinants of the equity risk premium. Under i.i.d. assessments the law of large numbers imply a single social state of belief and we show that the RBE of such economies have the same number of prices as in rational expectations equilibrium (REE). However, the RBE may exhibit large fluctuations if agents are allowed to hold extreme beliefs. Establishing 5% boundary restrictions on beliefs we show that the model with a single social state of belief cannot explain all the moments of the observed distribution of returns. The introduction of correlation among the beliefs of agents leads to the creation of new social states. We then show that under such correlation the model simulations reproduce the values of four key moments of the empirical distribution of returns. The observed equity premium is then explained by two factors. First, investors demand a higher risk premium to compensate them for the endogenous increase in the volatility of returns. Second, at any moment of time there are in the markets both rational optimists as well as rational pessimists and such a distribution leads automatically to a decrease in the riskless rate and to an increase of the risk premium. Correlation among beliefs of agents leads to fluctuations over time in the social distribution of beliefs and such fluctuations cause a higher equilibrium equity risk premium.