Endogenous Uncertainty and Market Volatility

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 price\dividend 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/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.

Risk Premia, diverse belief and beauty contests

We present a theoretical and empirical evaluation of the role of market belief in the structure of risk premia. To that end we employ a familiar asset pricing model for which we develop in detail the belief structure. The novelty in this development is the treatment of individual and market beliefs as Markov state variables. Moreover, the market belief is observable and the paper explains how we extract it from the data. The advantage of our formulation is that it permits a closed form solution of equilibrium prices hence we can trace the exact effect of market belief on the time variability of equilibrium risk premia. We present a model of asset pricing with diverse beliefs. We then explore the conditions under which diverse beliefs arise. We then derive the equilibrium asset pricing and the risk premium which the model implies. Since asset prices are affected by the dynamics of market belief, the component of market risk which is determined by the belief of agents is thus termed “Endogenous Uncertainty.” The theoretical conclusions are tested empirically for investments in the futures markets, the bond markets. Our main theoretical and empirical result is that fluctuations in the market belief about state variables are a dominant factor determining the time variability of risk premia. More specifically, we show that when the market holds abnormally favorable belief about future payoffs of an asset the market views the long position as less risky and hence the risk premium on that asset declines. This means that fluctuations in risk premia are inversely related to the degree of market optimism about future prospects of asset payoffs. This effect is very strong and empirically very dominant. The strong effect of market belief on market risk premia offers two additional perspectives. First, it offers an additional way of showing (for those who have any doubt) that fundamental factors affect market dynamics but perceptions have equally important effect on volatility. Second, that market belief is actually an observable data which can be used for a deeper understanding of the basic causes of stochastic volatility and time variability of risk premia.Risk premium; heterogenous beliefs; market state of belief; asset pricing; Bayesian learning; updating beliefs; Rational Beliefs

Contracting with Risk Aversion and Subjective Beliefs Under Costly State Verification

We study the loan contracing problem of Gale and Hellwig (1985) under general assumptions of risk aversion and possibly diverse subjective beliefs of the borrower and lender about the income of the investment. We characterize the optimal contract and show that (i) the contractual payoff in verification states varies by states in accord with risk aversion and probability belief of the borrower and lender, and (ii) teh verification region may consist of many intervals. Under these general assumptions, verification states are not necessarily interpreted as "default" states. Rather, they also reflect the need of the parties to trade on their differences in probability in the absence of markets for contingent claimsFinancial contract, costly state verification, risk aversion, subjective expectations

The role of expectations in economic fluctuations and the efficacy of monetary policy

We show diverse beliefs is an important propagation mechanism of fluctuations, money non neutrality and efficacy of monetary policy. Since expectations affect demand, our theory shows economic fluctuations are mostly driven by varying demand not supply shocks. Using a competitive model with flexible prices in which agents hold Rational Belief (see Kurz (1994)) we show that (i) our economy replicates well the empirical record of fluctuations in the U.S. (ii) Under monetary rules without discretion, monetary policy has a strong stabilization effect and an aggressive anti-inflationary policy can reduce inflation volatility to zero. (iii) The statistical Phillips Curve changes substantially with policy instruments and activist policy rules render it vertical. (iv) Although prices are flexible, money shocks result in less than proportional changes in inflation hence the aggregate price level appears "sticky" with respect to money shocks. (v) Discretion in monetary policy adds a random element to policy and increases volatility. The impact of discretion on the efficacy of policy depends upon the structure of market beliefs about future discretionary decisions. We study two rationalizable beliefs. In one case, market beliefs weaken the effect of policy and in the second, beliefs bolster policy outcomes and discretion could be a desirable attribute of the policy rule. Since the central bank does not know any more than the private sector, real social gain from discretion arise only in extraordinary cases. Hence, the weight of the argument leads us to conclude that bank´s policy should be transparent and abandon discretion except for rare and unusual circumstances. (vi) An implication of our model suggests the current effective policy is only mildly activist and aims mostly to target inflation

A Strategic Theory of Inflation

A strategic mechanism of price adjustment is introduced to explain inflations in the U.S. during 1909-1974. The mechanism follows from our theory that when the profit rate is above a normal-target rate, competitive forces operate to lower prices while if the profit rate is below the target a correlated strategy among firms operates to generate a rise in prices as a strategy to improve profitability. The notion of "correlated strategy" is adopted from game theory. The mechanism may operate in harmony or against demand and the net effect is what we call the "basic inflation." Contrary to a-priori notions of positive association between inflation rates and profit rates, our theory proposes a critical test of a negative association between these variables. Such a relationship is in fact empirically established. The analysis shows that large and persistent inflationary pressures are generated by low profitability and during 1971-1977 those accounted for some 20%-50% of total inflation. These pressures would be present even if no increase in cost occurs. This suggests that an important cause of the 1970's inflation is the low profit rate in the private sector and any public policy against inflation will fail if it does not aim at the same time to raise the profit rate on private capital.

The Role of Expectations in Economic Fluctuations and the Efficacy of Monetary Policy

We show diverse beliefs is an important propagation mechanism of fluctuations, money non neutrality and efficacy of monetary policy. Since expectations affect demand, our theory shows economic fluctuations are mostly driven by varying demand not supply shocks. Using a competitive model with flexible prices in which agents hold Rational Belief (see Kurz (1994)) we show that (i) our economy replicates well the empirical record of fluctuations in the U.S. (ii) Under monetary rules without discretion, monetary policy has a strong stabilization effect and an aggressive anti-inflationary policy can reduce inflation volatility to zero. (iii) The statistical Phillips Curve changes substantially with policy instruments and activist policy rules render it vertical. (iv) Although prices are flexible, money shocks result in less than proportional changes in inflation hence the aggregate price level appears “sticky” with respect to money shocks. (v) Discretion in monetary policy adds a random element to policy and increases volatility. The impact of discretion on the efficacy of policy depends upon the structure of market beliefs about future discretionary decisions. We study two rationalizable beliefs. In one case, market beliefs weaken the effect of policy and in the second, beliefs bolster policy outcomes and discretion could be a desirable attribute of the policy rule. Since the central bank does not know any more than the private sector, real social gain from discretion arise only in extraordinary cases. Hence, the weight of the argument leads us to conclude that bank’s policy should be transparent and abandon discretion except for rare and unusual circumstances. (vi) An implication of our model suggests the current effective policy is only mildly activist and aims mostly to target inflation.Monetary policy rules, Money non neutrality, Business cycles, Market volatility, Propagation mechanism, Capacity utilization, Heterogenous beliefs, Over confidence, Rational Belief, Optimism, Pessimism, Non stationarity, Empirical distribution

Endogenous uncertainty: A unified view of market volatility

The purpose of this paper is to provide a non-technical exposition of the main conclusions of the theory of Rational Belief Equilibrium (RBE) for market volatility. It is argued that the theory of Rational Belief Equilibria (RBE) provides a unified paradigm for explaining market volatility by the effect of "Endogenous Uncertainty" on financial markets. This uncertainty is propagated within the economy (hence "endogenous") by the beliefs of the agents who trade assets. The theory of RBE was developed in a sequence of papers assembled in a recently published book (See Kurz [1997]) and the present paper provides a non-mathematical exposition of both the main ideas of the theory of RBE as well as a summary of the main results of the book regarding market volatility. The structure of the paper is as follows. Section I outlines the basic assumptions underlying models of rational expectations equilibria (REE) and their implications to the study of market volatility. The paper reviews four basic problems which have constituted puzzles or anomalies in REE : (i) Why are asset prices much more volatile than their underlying fundamentals? (ii) The equity premium puzzle: why under REE the predicted riskless rate is so high and the equity risk premium so low? (iii) Why do asset prices exhibit the "GARCH" behaviour without exogenous fundamental variables to explain it? (iv) the "Forward Discount Bias" in foreign exchange markets: why are interest rate differentials such poor predictors of future changes in the exchange rates? Section II outlines the basic ideas and assumptions of the theory of RBE and the main proposition which it implies in relation to the problems of market volatility. Section III first develops the simulation models of RBE which are used in the analysis of the four problems above and explains that the domestic economy is calibrated, as in Mehra and Prescott [1985], to the U.S. economy. Then for each of the four problems the relevant simulation results are presented and compared both to the results predicted by a corresponding REE as well as to the actual empirical observations in the U.S. An Appendix reviews the results of additional econometric studies which bear on the results presented in the main text. The conclusion of the paper is that the main cause of market volatility is the distribution of beliefs and expectations of agents. The theory of RBE shows that if agents disagree then the state of belief of each agent, represented by his conditional probability, must fluctuate over time. Hence the nature of the distribution of the individual states of belief in the market is the root cause of all phenomena of market volatility. The paper shows that the GARCH phenomenon of time varying variance of asset prices is explained in the simulation model by the presence of both persistence in the states of beliefs of agents as well as correlation among the beliefs of the different agents. Correlation makes beliefs either narrowly distributed (i.e. "consensus") or widely distributed (i.e. "non-consensus"). When a belief regime of consensus is established (and due to persistence it remains in place for a while) then agents seek to buy or sell the same portfolio leading to high volatility. On the other hand, the widespread disagreement in a belief regime of non-consensus leads to balance between sellers and buyers leading to low market volatility. In short, the theory proposes that the GARCH phenomenon is the result of shifts in the distribution of beliefs in the market and these shifts are caused by the dynamics and correlation among beliefs of the agents. Analysis of the equity risk premium shows that the key question is what are the conditions on beliefs which will ensure that the average riskless rate is low and hence the average equity risk premium is high. It turns out that the key condition requires that the impact of the pessimists dominate the market a significant fraction of time. When this occurs they protect their endowment by shorting the stock and increasing their purchases of the safe riskless bill. This tends to bid up the price of the bill and lowers the price of the stock resulting in a lower riskless rate and a higher equity risk premium. The simulation results also show that correlation among the beliefs of the agents can change the frequency at which prices are realised over time and this implies that the correlation can increase the equity premium by increasing the frequency of realisation of those prices in which the pessimists have the greater impact on equilibrium prices

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

Beauty contests under private information and diverse beliefs: how different?

Abstract: The paper contrasts theories that explain diverse belief by asymmetric private information (in short PI) with theories which postulate agents use subjective heterogenous beliefs (in short HB). We focus on problems where agents forecast aggregates such as profit rate of the S&P500 and our model is similar to the one used in the literature on asset pricing (e.g. Brown and Jennings (1989), Grundy and McNichols (1989), Allen, Morris and Shin (2003)). We first argue there is no a-priori conceptual basis to assuming PI about economic aggregates. Since PI is not observed, models with PI offer no testable hypotheses, making it possible to prove anything with PI. In contrast, agents with HB reveal their forecasts hence data on market belief is used to test hypotheses of HB. We show the common knowledge assumptions of the PI theory are implausible. The theories differ on four main analytical issues. (1) The pricing theory under PI implies prices have infinite memory and at each t depend upon unobservable variables. In contrast, under HB prices have finite memory and depend only upon observable variables. (2) The “Beauty Contest” implications of the two are different. Under PI today’s price depends upon today’s market belief about tomorrow’s mean belief about “fundamental” variables. Under HB it depends upon today’s market belief about tomorrow’s market beliefs. Tomorrow’s beliefs are, in part, beliefs about future beliefs and are often mistaken. Market forecast mistakes are key to Beauty Contests, and are a central cause of market uncertainty called “endogenous uncertainty.” (3) Contrary to PI, theories with HB have wide empirical implications which are testable with available data. (4) PI theories assume unobserved data and hence do not restrict behavior, while rationality conditions impose restrictions on any HB theory. We explain the tight restrictions on the model’s parameters imposed by the theory of Rational Beliefs