# Endogenous Uncertainty and Market Volatility

01.01.1999

Mordecai Kurz, Joan Kenney, Maurizio Motolese

D5,D84,G12

Rational Expectations,Rational Beliefs,Rational Belief Equilibrium (RBE),Endogenous uncertainty,States of belief,Stock price,Discount bond,Equity premium,Market volatility,GARCH,Forward Discount Bias

Economy and Society

Fausto Panunzi

Endogenous Uncertainty is that component of economic risk and market volatility which is propagated within the economy by the beliefs and actions of agents. The theory of Rational Belief (see Kurz [1994]) permits rational agents to hold diverse beliefs and consequently, a Rational Belief Equilibrium (in short, RBE) may exhibit diverse patterns of Endogenous Uncertainty. This paper shows that most of the observed volatility in financial markets is generated by the beliefs of the agents and the diverse market puzzles which are examined in this paper, such as the equity premium puzzle, are all driven by the structure of market expectations. To make the case for this theory we present a single RBE model, which builds on developments in Kurz and Beltratti [1997] and Kurz and Schneider [1996], with which we study a list of phenomena that have been viewed as "anomalies" in financial markets. The model is able to predict the correct order of magnitude of:

(i) the long term mean and standard deviation of the pricedividend ratio;

(ii) the long term mean and standard deviation of the risky rate of return on equities;

(iii) the long term mean and standard deviation of the riskless rate;

(iv) the long term mean equity premium.

In addition, the model predicts:

(v) the GARCH property of risky asset returns;

(vi) the Forward Discount Bias in foreign exchange markets.

We also conjecture that an adaptation of the same model to markets with derivative assets will predict the appearance of "smile curves" in derivative prices.

The common economic explanation for these phenomena is the existence of heterogeneous agents with diverse but correlated beliefs. Given such diversity, some agents are optimistic and some pessimistic. We develop a simple model which allows agents to be in these two states of belief but the identity of the optimists and the pessimists fluctuates over time since at any date any agent may be in these two states of belief. In this model there is a unique parameterisation under which the model makes all the above predictions simultaneously. That is, although the parameter space of the RBE is large, all parameterisations outside a small neighbourhood of the parameter space fail significantly to reproduce some subset of variables under consideration. Any parameter choice in this small neighbourhood requires the optimists to be in the majority but the rationality of belief conditions of the RBE require the pessimists to have a higher intensity level. This higher intensity has a decisive effect on the market: it increases the demand for riskless assets, decreases the equilibrium riskless rate and increases the equity premium. In simple terms, the large equity premium and the lower equilibrium riskless rate are the result of the fact that at any moment of time there are agents who hold extreme pessimistic beliefs and they have a relatively stronger impact on the market. The relative impact of these two groups of agents who are, at any moment of time, in the two states of belief is a direct consequence of the rationality of belief conditions and in that sense it is unique to an RBE.

As for the correlation among the beliefs of agents, the paper shows that the dynamics of asset prices are strongly affected by such correlation. The pattern of correlation which was used in the model can be explained intuitively in terms of its effect on the dynamics of prices. The model correlation causes periods of price rises (i.e. bull markets) to develop slower than periods of decline (i.e. bear markets) hence the model dynamics does not permit prices to shoot directly from the bottom to the top but the opposite is possible and takes the form of market crashes.

Note: Both the RBE model developed in this paper as well as the associated programs used to solve it are available to the public on Professor Kurz’s web page at http://www.stanford.edu/~mordecai/