Rational Expectations, Risk and Chaos in Financial Markets

Abstract

Begg (1984) has recently presented a rational expectations model of equilibrium bond pricing. One important feature of this model is that lenders making portfolio choices between short and long assets are risk-averse, so that the well-known principle of certainty equivalence no longer applies to the optimal demand for bonds. When the policy for the supply of bonds and a simple policy rule for interest rates are specified, an equilibrium model of equilibrium bond pricing is obtained where expectations are self-fulfilling in both the mean and variance. The key feature of this interesting model of the bonds market is that it is nonlinear. As long as the degree of risk, the degree of risk-aversion or the supply of bonds are not too large, the forward evolution of the model will be unstable and a unique rational expectations solution exists as the backward evolution of equilibrium bond prices will converge. This is very much in the spirit of the conventional saddlepoint approach to solving rational expectations models (Blanchard and Kahn, 1980), where equilibrium asset prices are considered as forward-looking jump variables. However, if the degree of risk, the degree of risk-aversion, the coupon or the supply of bonds become large enough, the forward evolution of the model becomes locally stable. Begg's (1984) interpretation is that this leads to a loss of confidence and financial panic, because there is now an infinite number of convergent rational expectations trajectories. The main objective of this paper is to show that this interpretation is incomplete and to formalise Begg's notion of financial panic. Because the forward evolution of asset prices is stable, the backward evolution is unstable when there is a large degree of risk or riskaversion. It therefore seems as if equilibrium bond prices have become predetermined and are no longer jump variables, so that it seems as if risk-aversion investors have 'given up' their forward-looking behaviour in the face of too many shocks. This is only a locally valid argument and contradicts intuition, because efficient asset prices are generally regarded as jump variables that incorporate 'news' about current and future events. It is possible, however, still to have forward-looking asset prices, because increasing the degree of uncertainty or the degree of risk-aversion eventually leads to unique speculative bubbles and possibly financial chaos for the backward evolution of equilibrium bond prices. There exists therefore, in addition to the infinity of convergent and pre-determined rational expectations paths, a unique non-