### Determinants of Stock Market Volatility and Risk Premia

"... Summary. We show the dynamics of diverse beliefs is the primary propagation mechanism of volatility in asset markets. Hence, we treat the characteristics of the market beliefs as a primary, primitive, explanation of market volatility. We study an economy with stock and riskless bond markets and for ..."

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Summary. We show the dynamics of diverse beliefs is the primary propagation mechanism of volatility in asset markets. Hence, we treat the characteristics of the market beliefs as a primary, primitive, explanation of market volatility. We study an economy with stock and riskless bond markets and formulate a financial equilibrium model with diverse and time varying beliefs. Agents' states of belief play a key role in the market, requiring an endogenous expansion of the state space. To forecast prices agents must forecast market states of belief which are beliefs of "others" hence our equilibrium embodies the Keynes "Beauty Contest." A "market state of belief" is a vector which uniquely identifies the distribution of conditional probabilities of agents. Restricting beliefs to satisfy the rationality principle of Rational Belief (see Keywords: Market states of beliefs; market volatility; equity risk premium; riskless rate; over confidence; Heterogenous beliefs; Rational Belief; optimism; pessimism; empirical distribution. JEL Classification Numbers: G1; G12; E43; E44; D58; D84. Introduction The forces which determine equilibrium market volatility and risk premia are probably the most debated topics in the analysis of financial markets. The debate is driven, in part, by empirical evidence of market "anomalies" which have challenged students of the subject. Consumptionbased asset pricing theory has had a profound impact on our view of financial markets. Early work of Difficult to account for risk premia are not confined to the consumption based asset pricing theory; they arise in many asset pricing models. Three examples will illustrate. The Expectations Hypothesis is rejected by most studies of the term structure (e.g. Backus, We suggest market volatility and risk premia are primarily determined by the structure of agents' expectations called "market state of belief." Diversity and dynamics of beliefs are then the root cause of price volatility and the key factor explaining risk premia. Agents may be "bulls" or "bears." A bull at date t expects the date t+1 rate of return on investments to be higher than normal, where "normal" is defined by the empirical distribution of past returns. Date t bears expect returns at t+1 to be lower than normal. Agents do not hold Rational Expectations (in short RE) since the environment is dynamically changing, non stationary, and true probabilities are unknown to anyone. In such complex environment agents use subjective models. Some consider these agents irrational, but one cannot require them to know what they cannot know: there is a wide gulf between an RE agent and irrational behavior. We explore the structure of economies with diverse beliefs and show they must have an expanded state space. Our computing models assume agents hold rational beliefs in accord with the theory of Rational Belief Equilibrium (in short, RBE) due to The Rational Belief Principle. "Rational Belief" (in short, RB) is not a theory that demonstrates rational agents should adopt any particular belief. Indeed, since the RB theory explains the observed belief heterogeneity, it would be a contradiction to propose that a particular belief is the "correct" belief agents should adopt. The RB theory starts by observing that the true stochastic law 3 of motion of the economy is non-stationary with structural breaks and complex dynamics hence the probability law of the process is not known. Agents have a long history of past data generated by the process which they use to compute relative frequencies of finite dimensional events hence all finite moments. With this knowledge they compute the empirical distribution and use it to construct an empirical probability measure over sequences. In contrast with a Rational Expectations Equilibrium (in short, REE) where the true law of motion is known, agents in an RBE who do not know the truth, form subjective beliefs based only on observed data. Hence, any principle on the basis of which agents can be judged as rational must be based only on the data. Since a "belief" is a model of the economy together with a probability Earlier papers using the RBE rationality principle have also argued that agents' beliefs are central to explaining market volatility (e.g. Motolese The Main Results . First, "Belief states" are developed as a tool for equilibria with diverse belief. Next, our method is to use properties of market beliefs as primitive explanation of volatility. Two characteristics of beliefs fully account for all features of volatility and premia observed in markets: intensity of fat tails in the belief densities of agents; (B) asymmetry in the proportion of bull and bear states in the market over time. High intensity means agents exhibit rationalizable over confidence with fat tails in their subjective densities. Asymmetry in frequency of belief is a characteristic which says that on average, at more than half of the time agents do not expect to make excess returns. Our model also implies market returns must exhibit stochastic volatility which is generated by the dynamics of market belief. The Economic Environment The economy has two types of agents and a large number of identical agents within each type. An agent is a member of one of the two types of infinitely lived dynasties identified by their endowment , utility ( defined over consumption) and by their belief. A dynasty member lives a fixed short life and during his life makes decisions based on his own belief without knowing the states of belief of his predecessors. He is replaced by an identical member. There are two assets: a stock and a riskless, one period, bond. There is an aggregate output process which is Y t , t ' 1, 2, . . . divided between dividends paid to owners of the common stocks and non -D t , t ' 1, 2, . . . dividend endowment which is paid to the agents. The dividend process is described by 1 A Stable Process is defined in where { , t = 1, 2, ...} is a stochastic process under a true probability which is non-stationary with x t structural breaks and time dependent distribution. This time varying probability is not known by any agent and is not specified. Instead, we assume { , t = 1, 2, ...} is a stable process 1 hence it has x t an empirical distribution which is known to all agents who learn it from the data. This empirical distribution is represented by a stationary Markov process, with a year as a unit of time, defined by 2 (2) with x ) The infinitely lived agents are enumerated j = 1, 2 and we use the following notation: -consumption of j at t; C j t -amount of stock purchased by j at t; θ j t -amount of one period bond purchased at discount by agent j at t; B j t -stock price at date t; q s t -the discount price of a one period bond at t; q b t -non capital income of agent j at date t; Λ j t H t -information at t, recording the history of all observables up to t. Given probability belief , agent j selects portfolio and consumption plans to solve the problem Q j t (3a) Max We assume additively separable, power utility over consumption, a model that failed to generate premia in other studies (see, Campbell and Cochrane (2000) The Euler equations are and the market clearing conditions are then A Rational Expectations Equilibrium (REE) Strictly speaking we cannot evaluate the REE since the true output process has not been specified. We thus define an REE to be the economy in which all agents believe that 4 when C t approaches the mean of past consumption? Campbell and Cochrane (1999, page 244) show that for the model to generate the desired moments, the degree of risk aversion is 80 at steady state and exceeds 300 frequently along any time path. If instead we use Abel's (1990), (1999) formulation , marginal utility is normalized to be 1 at habit but the model does not generate volatility. Second, big fluctuations of stock prices are observed during long periods when consumption grows smoothly as was the case during the volatile period of 1992 -2002. Finally, the model predicts perfect correlation between consumption growth and stock returns but the record (see 7 price\dividend ratio , (ii) the risky return R, (iii) the riskless rate ; the equity premium , the q s e p Sharp Ratio and the correlation between x and R. Moments of market data vary with sources reporting and methods of estimation. The market data reported in The "Equity Premium" is not the only puzzle; the wider question is how to explain market volatility. The General Structure of Equilibria with Diverse and Time Dependent Beliefs Although Our methodology is to use the distribution of beliefs to explain market volatility hence we need to determine a level of detail at which agents "justify" their beliefs. If we aim at a complete specification of such modeling, our study is doomed to be bogged down in details of inference from small samples and information processing. Although interesting, from a general equilibrium perspective it is not needed. To study volatility we focus on a narrow but operational question. Since (4a)-(4b) require specification of conditional probabilities, we need only a tractable way to describe differences among agents' beliefs and time variability of their conditional probabilities, without fully specified models to justify them. From our point of view what matters is the fact that market beliefs are diverse and time dependent; the reasoning which lead agents to the subjective models are secondary. The tool we developed for this goal is the individual and the market "state of belief" which we now explain. Market States of Belief and Anonymity: Expansion of the State Space The usual state space for agent j is denoted by S j but when beliefs change over time we introduce an additional state variable called "agent j state of belief ." It is a variable generated by agent j, expressing his date t subjective view of the future and denoted by . It has the g j t 0 G j property that once specified, the conditional probability function of an agent is uniquely specified and 5 Note that larger values of g imply a more bearish perspective. In the applications below larger g will express an agent's reduced probability belief in making excess returns. This may appear unnatural but we study equilibria with asymmetry measured by the frequency at which an agent is a bull or a bear. One of our main results says that the data supports a model where, on average, agents expect to make excess returns less than 50% of the time or, equivalently, that in a large market a majority of agents are pessimistic about making excess returns. We thus focus on the market pessimists. Since in the computational model we use a logistic function to express this asymmetry, it turns out that the use of a logistic function necessitates the condition that a larger g means more bearishness. Without the desired asymmetry g could have an opposite interpretation. We discuss this point further in Appendix A. is actually a proxy for j's conditional probability function. We note g j t that are privately perceived by agent j and have meaning only to him. Since a dynasty consists of g j t a sequence of decision makers, used by j has no impact on the description of beliefs by other g j t dynasty members. In the model of this paper agents forecast dividend or profit growth rate x t+1 (i.e. the exogenous variable) hence describes agent j conditional probability of profit growth at g j t 0 ú t+1. We shall permit rational agents to be "bulls" who are optimistic about future excess returns or "bears" who are pessimistic about future excess returns. To understand the role of we introduce g he is a bull and makes g j t < a higher profit growth forecasts than the ones implied by As indicated, we do not explain the reasoning used by agents to deviate from the empirical forecast. It is a common practice among forecasters to use the strict econometric forecast only as a benchmark. Given such benchmark, a forecaster uses his own model to add a component reflecting an evaluation of circumstances at a date t that call for a deviation at t from the benchmark. In short, is a description of how the model of agent j deviates from the statistical forecast implied by (2). g j t In this paper we assume that at any date the state of belief is a realization of a process of the form Persistent states of belief which depends upon current market data fit different cases of economies with diverse beliefs. We consider three examples to illustrate how one may think about them. agents to assume they cannot affect endogenous variables. It is analogous to requiring a competitive firm to assume it has no effect on prices. The issue here is the specification of how agents forecast prices. To that end we define the "market state of belief " as a vector , keeping in z t 11 is so central to our approach that we use three notational devices to highlight it: We now return to the economy with two agent types and simplify by assuming the market belief is observable. This assumption is entirely reasonable since there is a vast amount of (z 1 t , z 2 t ) public data on the distribution of forecasts in the market and on the dynamics of this distribution. Indeed, using forecast data obtained from the Blue Chip Economic Indicators and the Survey of Professional Forecasters we constructed various measures of market states of belief. Since (z 1 t , z 2 t ) is observable we need to modify the empirical distribution (2) and include in it. We assume (z In any application one assumes the parameters of We then estimated principal components to handle multitude of forecasted variables (for details, see Fan % Σ Finally, denote by m the probability measure on infinite sequences implied by (8) with the invariant distribution as the initial distribution. We then write where H t is the history at t. E m ( w t%1 | H t )

### by

, 2007

"... economic and public policy issues. The SIEPR Discussion Paper Series reports on research and policy ..."

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economic and public policy issues. The SIEPR Discussion Paper Series reports on research and policy

### SIEPR Discussion Paper No. 0602 Beauty Contests Under Private Information and Diverse Beliefs: How Different? By

, 2006

"... 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 o ..."

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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 (2006)). 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 which is correct and depends upon supply shock and inference from prices. 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

### Consensus Investor and Intertemporal Asset Pricing with Heterogeneous Beliefs ∗

"... This paper considers an intertemporal capital market with an arbitrary number of heterogeneous belief investors who maximise the expected utilities of their terminal wealths under their subject beliefs. By aggregating the heterogeneous beliefs of the investors, a consensus investor is constructed to ..."

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This paper considers an intertemporal capital market with an arbitrary number of heterogeneous belief investors who maximise the expected utilities of their terminal wealths under their subject beliefs. By aggregating the heterogeneous beliefs of the investors, a consensus investor is constructed to characterise the market equilibrium under the heterogeneous beliefs. Applying this idea to logarithmic utility function case, the impact of the belief heterogeneity on the market equilibrium is examined explicitly and survival conditions of the heterogeneous agents are given.

### The Role of Diverse Beliefs in Asset Pricing and Equity Premia 1

"... Why do risk premia vary over time? We examine this problem theoretically and empirically by studying the effect of market belief on risk premia. Individual belief is taken as a fundamental, primitive, state variable. Market belief which is the distribution of individual beliefs is observable, it is ..."

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Why do risk premia vary over time? We examine this problem theoretically and empirically by studying the effect of market belief on risk premia. Individual belief is taken as a fundamental, primitive, state variable. Market belief which is the distribution of individual beliefs is observable, it is central to the empirical evaluation and we show how to measure it. Our asset pricing model is familiar from the noisy REE literature but we adapt it to an economy with diverse beliefs. We derive equilibrium asset prices and implied risk premium. Our approach permits a closed form solution of prices hence we trace the exact effect of market belief on the time variability of asset prices and risk premia. We test empirically the theoretical conclusions. We study premia on long positions of S&P500 for investment periods of 6-12 months and find that, on average, a change in the mean market belief by 1 standard deviation changes equity premia by about 0.4 standard deviation of excess returns. We also find that the introduction of belief variables reduces the effect of the traditional Fama and French (1989) variables used to forecast excess returns. More specifically we find that given information about market belief the term premium, the default premium and Lattau and Ludvigson’s (2001) CAY variables have no significant effect. As to the structure of the premium we show that when the market holds abnormally favorable belief about the future payoff on the S&P500 the market views the long position as less risky hence the equity risk premium declines. We also find that in a VAR model of asset and consumption growth, our market belief variables have a significant effect. We thus conclude that market belief has significant independent effect on economic fluctuations and risk premia.

### Prepared for the Handbook of Finance Series Volume Entitled Handbook of Financial Markets: Dynamics and Evolution

, 2008

"... Abstract: This is a chapter in a new series of Handbooks in Financial Economics entitled “Dynamics and Evolution of Financial Markets. ” It explores works which contribute to explaining market dynamics and volatility by employing models of rational but diverse beliefs. This excludes models of Behavi ..."

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Abstract: This is a chapter in a new series of Handbooks in Financial Economics entitled “Dynamics and Evolution of Financial Markets. ” It explores works which contribute to explaining market dynamics and volatility by employing models of rational but diverse beliefs. This excludes models of Behavioral Economics. We explore models that permit endogenous amplification which can, in turn, explain such basic facts as excess volatility and high risk premia. The first part gives an exposition of the Noisy Rational Expectations (i.e. REE) Asset Pricing Theory where diversity of beliefs arises from diverse private information. Examination of these models enable us to evaluate the assumptions made and the ability of the models to explain the data on market dynamics. Our conclusion is that, although Noisy REE asset pricing theories have many attractive features, they do not offer a satisfactory paradigm for market dynamics. Indeed, for most models increased amount of idiosyncratic private information leads to a reduction in volatility. We next turn to models of diverse beliefs without private information and explore work on Rational Belief Equilibria starting with modeling beliefs. Agents do not know the structure of the dynamically complex economy but have data over a long time to enable the computation of an empirical probability on sequences which is then commonly known. A belief is a model of deviations from the empirical probability. It is shown that in a wold of diverse beliefs, to be rational a belief must fluctuate, generating a mechanism of endogenous amplification of market dynamics. An important innovation of the theory is the treatment of individual beliefs as state variables and market belief is then derived as the distribution of individual beliefs. Market belief is observable via sampling data from several sources and we review both data sources as well as computations of the first two moments of market belief. We explore an

### MPRA Munich Personal RePEc Archive

, 2006

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

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

### FOR MAY 12-14 UCI CONFERENCE ON MATHEMATICAL ECONOMICS

, 2006

"... Abstract: The paper contrasts implications of theories which explain diverse belief by asymmetric private information (in short PI) with theories which postulate agents use heterogenous probability models (in short HB). We focus on the use of such theories in solving problems where agents forecast e ..."

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Abstract: The paper contrasts implications of theories which explain diverse belief by asymmetric private information (in short PI) with theories which postulate agents use heterogenous probability models (in short HB). We focus on the use of such theories in solving problems where agents forecast economic aggregates such as the S&P500, growth rate of GDP, inflation rates, etc. We study implications of the two asset pricing theories with a model where each agents trades for two periods. Such a model has been used extensively in the literature on asset pricing (e.g. Brown and Jennings (1989), Grundy and McNichols (1989), Allen, Morris and Shin (2005)). On conceptual grounds we find that there is no basis to assuming PI about economic aggregates. In contrast, HB theories are naturally suited. Since PI is not observed, models with PI offer little in the way of testable hypotheses, making it possible to prove anything with PI. In contrast, agents who hold HB are happy to reveal their forecasts hence there is vast data on market belief which can be used to test hypotheses about the effect of diversity. The two asset pricing models differ on several central issues. (i) Models with PI introduce artificial “noise” such as unobserved shocks to the supply of securities to explain diversity of forecasts but diversity does not persist. It vanishes as the number of trading rounds increases. In contrast, HB theories have a natural renewal mechanism to explain the perpetual persistence of diversity; (ii) under both theories the market belief fails iterated expectations but, in economic substance, the Beauty Contest implications of the two are different. Under PI today’s price depends upon

### CENTRE FOR DYNAMIC MACROECONOMIC ANALYSIS WORKING PAPER SERIES CDMA10/10 Learning about Risk and Return: A Simple Model of Bubbles and Crashes *

, 2010

"... This paper demonstrates that an asset pricing model with least-squares learning can lead to bubbles and crashes as endogenous responses to the fundamentals driving asset prices. When agents are risk-averse they need to make forecasts of the conditional variance of a stock’s return. Recursive updatin ..."

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This paper demonstrates that an asset pricing model with least-squares learning can lead to bubbles and crashes as endogenous responses to the fundamentals driving asset prices. When agents are risk-averse they need to make forecasts of the conditional variance of a stock’s return. Recursive updating of both the conditional variance and the expected return implies several mechanisms through which learning impacts stock prices. Extended periods of excess volatility, bubbles and crashes arise with a frequency that depends on the extent to which past data is discounted. A central role is played by changes over time in agents ’ estimates of risk.

### Defaults, Shortsales and the Social Costs of Volatility

, 2009

"... This paper examines the welfare effects of credit and shortsales constraints and limited liability/minimum consumption guarantee in an overlapping generations (OLG) model with rational beliefs in the sense of Kurz (1994). To measure the social welfare, it instead adopts an ex post social welfare con ..."

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This paper examines the welfare effects of credit and shortsales constraints and limited liability/minimum consumption guarantee in an overlapping generations (OLG) model with rational beliefs in the sense of Kurz (1994). To measure the social welfare, it instead adopts an ex post social welfare concept in the sense of Hammond (1981), since the standard Pareto criterion becomes inappropriate when het-erogeneous beliefs are present. Simulation results indicate a trade-off between a larger opportunity set and a larger room for ‘mistakes’, and thus, the existence of a socially optimal level of various constraints. 1