Information Processing and Financial Market Price Adjustments

Using a model of heterogeneous investors’ responses 
to changing information, this paper studies the impact of learning on 
equilibrium price changes. The paper develops the comparative statics of single 
period equilibrium conditions. It characterizes generally the variety of 
circumstances encountered in the literature.


Introduction
Traditionally, finance theory has argued that securities prices are determined by publicly available fundamental (long run) information. However there are many conflicting findings. In this section we cite only closely related examples. More details are provided in the reference section. To cite selected examples, in the short run empirical estimates of risk neutral probabilities from option prices data display dynamic responses to changing market conditions (e.g. Figlewski [1]). In the longer run, observers attribute significant market events to differing causes. Campbell, Giglio and Polk [2] find that the price impacts of the 2000-2002 tech crash were due principally to changes in expectations (discount rates) while the initial price impacts of the 2007-2009 crash were due mainly to changes in fundamentals (cash flow forecasts). Moskowitz, Ooi and Pedersen [3] show that both time series momentum and value effects are commonly observed across nearly sixty different markets. In these markets, prices relative to fundamentals can remain unrepresentatively high-or low-for periods of up to a year. This paper ascribes observed price behavior to the evolution of heterogeneous expectations that follow on public information releases. Many information

Passive and Active Responses to Prices
In our single period model investors have heterogeneous expectations and equilibrium prices are established by a marginal investor. Non-marginal investors take these prices as given, defining an equilibrium with relatively straightforward properties. While the literature has not fully established the empirical nature of expectations change, Haltiwanger and Waldman [4] propose theoretical contexts in which the proportions of sophisticated to unsophisticated agents can have differential effects on equilibrium. Our model incorporates a spectrum of differently informed agents whose interactions determine equilibrium price adjustments.
The outline of the paper is as follows. Section 2 discusses relevant theoretical and empirical literature while Sections 3 and 4 present the single period equilibrium model. We determine equilibrium effects of changing probability estimates-assuming that investors agree on state-dependent payoffs. We can also determine comparative statics results for changes in payoffs.

Review of Literature
This section offers a sketch of research themes related to the paper's findings.
This section is selective and focuses on closely relevant papers.

Divergence of Opinion and Price Impacts
To Edward Miller [5], heterogeneous expectations equilibrium implies an optimistic minority will invest in a risky security. Miller further notes that expected returns on riskier securities might change in either direction as opinion diverges.
Our model shows that investor expectations of asset payoffs determine whether the optimists (called speculators and defined as purchasers of upstate claims) constitute a majority or a minority, and finds equilibria for both cases. Our model also identifies uniquely different conditions that imply increased returns under one set of conditions, decreased returns on another.

Dynamic Analyses
Other heterogeneous expectations models investigate equilibria in which investors respond immediately and strategically to an evolving environment. For example, Harrison and Kreps (HK) [6] show that equilibrium prices depend on how investors believe others will respond to their actions. HK [6] show further that if an equilibrium price exists in their model, it will exceed the price any investor would be willing to pay if obliged to hold the stock forever. In this context of reactive trading, heterogeneity increases potential profit opportunities because DeTemple and Murthy (DTM) [7] examine interest rates, asset prices, and asset holdings in an economy with heterogeneous and rationally updated beliefs.
The equilibrium interest rate becomes a weighted average of the rates that would prevail in economies with homogeneous agents who hold the beliefs of different agents in the heterogeneous model. The weights are fractions of total wealth held. In DTM [7] financial innovation affects both quantity and price dynamics.
Irrational investors are eventually bankrupted through trading with their rational counterparts, but only after a very long time; several hundred years in one example. Xiong and Yan (XY) [8] study dynamic equilibria in bond markets, assuming two groups of investors with different learning models. Since the groups are motivated to take speculative positions against each other, investor activity generates wealth fluctuations that increase asset price volatility and contribute to time variation in risk premia. By choosing particular parameters for their learning models, XY [8] isolate belief-dispersion effects from such other effects as erroneous average belief and underestimation of risk. Their work explains excessive volatility in bond yields, the failure of the expectations hypothesis, and the ability of a linear combination of forward rates to predict bond returns.
Neave [9] assumes investors are passive price-takers who use options to trade in an incomplete market. As options become closer substitutes for contingent claims, the incomplete markets equilibria converge smoothly to complete markets equilibria when investors behave as price takers. These results contrast with the Brock, Hommes, and Wagener [10] conclusion that hedging instruments can destabilize markets when traders react to each other.
This paper provides a theory capable of explaining both the traditional approach to valuation and departures from it. We assume heterogenous investors and the way their differences are resolved at equilibrium.

Selected Empirical Findings
Campbell, Giglio and Polk (CGP) [2] find that the stock market downturns of 2000-2002 and 2007-2009 have different proximate causes. CGP [2] attribute the early 2000s downturn to a large increase in the discount rates applied to profits by rational investors, while they attribute the late 2000s crash to a decrease in rational expectations of future profits. Our model provides a theoretical setting in which both discount rate and cash flow expectations can be identified and analyzed separately. Although we focus primarily on risk neutral investors, an Appendix finds a complete market equilibrium with risk averse investors. And the model can be extended further to distinguish between changes in discount rates and changes in expectations, although apart from an illustrative example we do not do so in this paper.
Asness, Moskowitz and Pedersen (AMP) [11]  that momentum loads positively on liquidity risk but negatively on value. AMP [11] suggest their results might be due to momentum in the most popular trades, as investors flock to assets whose prices have appreciated most recently. When a liquidity shock occurs, investors engaged in liquidating sell-offs will put more price pressure on the most popular and crowded trades. High momentum securities will be popular initially, but will also be securities from which everyone runs at the same time [12]. On the other hand, value purchases can represent a contrarian view that is less affected by the liquidity concerns surrounding the most popular assets. This paper's model can incorporate either liquidity or value effects, and different price patterns emerge according to investors' perception of an information event; i.e., the extent to which different investors interpret the same message differently.
Moskowitz, Ooi, and Pedersen (MOP) [3] document significant time series momentum in markets for 58 liquid securities. MOP [3] find persistence in returns over one to 12 months, and partial reversals over longer horizons. MOP [3] further find that while time series momentum strategies can generate substantial abnormal returns, such portfolios have little exposure to standard asset pricing factors. The MOP [3] momentum strategies perform best during extreme markets, and portfolio profits accrue primarily to speculators at the expense of hedgers. MOP [3] note that time series momentum matches predictions of both behavioral and rational asset pricing theories. Generally, behavioral theories envisage momentum as deriving from an initial underreaction to news followed by an upward drift and eventual overreaction, with undervaluation driven by conservatism and subsequent overvaluation based on representativeness (Barberis, Shleifer and Vishny [13]). The levels of these and other behavioral characteristics may be affected by investor sentiment (Baker and Wurgler [14]). Brown, Christensen, Elliott, and Mergenthaler [15] studied managers' financial reporting practices in relation to the BW sentiment index. On the other hand, MOP do not find sentiment effects in their data. Positive feedback trading is a related behavior-based theory of momentum, arising from self-attribution bias and overconfidence. This momentum is augmented as rational investors "jump on the bandwagon" to exploit a price runup while it lasts (DeLong, Shleifer, Summers, and Waldmann [16]). Price reversals eventuate when momentum collapses.
Rational theories usually view momentum as a reaction to changes in risk, and various risk proxies have also been studied. These include stochastic discount factor (Ahn, Conrad, and Dittmar [17]), dividend growth rate changes (Johnson [18]), firm revenue and cost volatility (Sagi and Seasholes [19]).
Although our model prices only two contingent claims, its structure provides insights into more broadly defined securities markets. We allow learning abilities to differ as primitives in our model, and explains momentum as a rational response of investors with different learning abilities and different strategies. It is in the rational camp since momentum can be interpreted as an initial reaction to

Model Structure
Our model adapts a structure developed by Fostel and Geanakoplous (FG) [20].
In a single period version, equilibrium is sought in a two-state complete markets setting with a continuum of rational risk-neutral investors. 1 In this and the following section, investors are assumed to agree on the sizes of state payoffs (fundamentals), but not on state probabilities (expectations). 2 Equilibrium prices are established by a marginal investor, and other investors trade at the marginal investor's price.

State Payoffs, Expectations, and Utility
Our single-period model considers times zero and one. At time zero the economy has a productive asset X that produces U units of a consumption good at time 1 if a high state occurs and D units in the low state, Each investor is endowed with both upstate and downstate contingent claims, upstate with density U and downstate with density D. Investors trade claims to maximize the expected utility of their endowment. Information and processing costs are assumed to be zero.
For simplicity, we assume the objective probabilities of realizing U and D are equal. 3 When for comparison purposes we postulate investors with homogeneous expectations, they will be assumed to use these equal probabilities. As to heterogeneous expectations, each investor for the realized state at t = 1. We further assume that ( ) U q h is a monotonically and strictly increasing function of h; hence To depict the impacts of changing expectations as simply as possible, we em- q h whose details will be specified shortly. We model a change in average investor expectations as a shift of the upstate probability function, and a change in agreement between investors as a change in the upstate function's slope.
Denote the investors' consumption levels at time 1 by C U and C D , where We later show that with risk averse investors the equilibrium has a similar qualitative structure. 2 We can also study changes in state payoffs, but apart from one example do not do so in this paper. 3 Asset prices in Arrow-Debreu equilibria depend only on subjective probabilities. But we need objective probabilities for comparison purposes; see Section 4.
We assume in attaining equilibrium, that all investors are risk-neutral and that each maximizes an expected utility: Thus claims are valued at time zero using the equilibrium prices U p and D p respectively; transactions costs are assumed to be zero. Purchases are restricted to the value of the cash raised; investors are not permitted to sell short.
At equilibrium, investor 1 h is defined as the marginal buyer who is indifferent between buying up or down claims at the prices U p and D p . Using (1) and (2), it is easy to verify that investors will buy only upstate claims, and in- only downstate claims. Consistent with prior literature, we call the former investors speculators, the latter hedgers. 5 The speculators' cash Since speculators and hedgers transact with each other, and since our market economy is assumed to be closed, equating revenues with expenditures for either class of traders defines the exchange of funds for both.

Equilibrium
Equilibrium is defined by investor proportions ( ) h is indifferent to buying the up or the down claim, the equilibrium must satisfy It is convenient to rewrite (5) as thus distinguishing cash constraint from expectations effects. To interpret the cash constraint side of (6), note that for any (3) and (4) are positive quantities. Hence we can take the ratio of (3) to (4) to write Empirically, hedgers are primarily large commercial firms while speculators are large investment houses. 6 The cash constraints represent aggregates of individual investors' budget constraints. Investors may repurchase some of the securities they first sell. FG [20] assume this trading pattern for analytical convenience, and we retain it, even though in the present context it is equally convenient to consider only net trading.
which implies proportionately more speculators than hedgers. The condition further implies that due to the influence of heterogeneous expectations, the equilibrium prices typically incorporate (positive or negative) risk premia. We write the expectations side of (6) as Then treating D and U as fixed, equating the expressions in (7) and (8) Henceforth, we denote the equilibrium claim price ratio using

Interpreting the Cash Constraints
For any can be rewritten cash constraints require that the ratio of speculators to hedgers equal the ratio of upside to downside expenditures on claims. Since (9) further implies that h and 1 -h can be interpreted as relative expenditure on hedging and speculation respectively. Since with risk neutral investors and zero time preference, equilibrium claim prices always satisfy the total value of securities expenditures increases with p U . In

Analyzing Myopic Equilibria
This section first shows that any change in Q, the ratio of investor expectations, defines an equilibrium change that is observationally distinct from one following on a change in P. The second subsection defines the rotations representing  Figure 1 graphs Q(h) and P(h) as defined by (7) and (8). The Figure uses generic properties of P(h) and Q(h) retained throughout the paper.

The Q and P Functions
Rewriting (8) as and recalling that Figure 1. Similarly, rewriting (7) as ( ) and treating U and D as parameters establishes though subsequent analyses will modify either or both Q and P, Q(h) will remain strictly increasing strictly convex, and P(h) strictly decreasing strictly convex throughout the paper.

Changes in Q and in P
Following an ambiguous information event, we interpret increasing agreement as a clockwise rotation of a linear upstate probability function. Investors may relocate themselves along the continuum. Given increasing agreement, some high state investors will lower their prior upstate probabilities, but not so much as to abandon their high state investment strategy. Other high state investors may switch to a low state investment strategy. Similarly, some low state investors may raise their prior high state probabilities sufficiently to switch to a high state investment strategy while others retain low state expectations. investors. In the rest of paper we shall study increasing agreement, in which case all rotations will be clockwise.

Information Processing and Changing Expectations
The graphs in this section outline our model and its evolving equilibrium prices.
Since there is a one-to-one mapping from rotations of q u (h) to those of Q u (h) we work with the latter in the graphs below. This means that Q 2 slopes less steeply than Q 1 , rotating clockwise around an intersection between Q 1 (h) and a horizontal, homogeneous expectations line (for clarity the latter is not shown). Under these assumptions, the new Q 2 (h) determines an equilibrium price ratio P e2 such that  We can also use Figure 3 to analyze the finding of CGP [2] that the price impacts

Conclusion
We have presented a  Figure 3 shows, if so, the two effects reinforce each other. We leave these matters to future papers.

Appendix: Equilibrium with Risk Averse Investors
This Appendix shows that in the complete market of this paper investor risk aversion will reduce both the upstate equilibrium price and speculators' purchases relative to their risk neutral values. The changes follow from a downward shift of the Q curve as established in this note, and depicted in the graph below.
(The effect in the graph is exaggerated to show it more clearly.) In addition, the magnitude of the downward shift can be assessed using the utility function's Arrow-Pratt index of risk aversion. Finally, we note that the model can distinguish the equilibrium effects of changes in risk aversion from changes in market risk.
Each investor h faces the expected utility maximization problem of allocating a fixed endowment between two contingent claims: where V is a utility function, C U is consumption in the upstate, C D in the downstate, F a fixed endowment 14 and p U , p D are equilibrium prices for contingent claims with payoffs U and D respectively. For any investor h the first order necessary conditions for C U and C D are: where λ is a Lagrange multiplier. Since all terms in Equation (A2.2) are positive, the equations' ratio can be taken to obtain At the risk neutral equilibrium ( ) ( ) ; .
is strictly monotone decreasing and the expectation of and her expected utility is ; .

D D q h V D q h V D h h > <
Just as in the risk-neutral environment, because of the monotonicity of their expectations functions non-marginal purchasers are satisfied with their positions and cannot improve them by changing their purchases at current equilibrium prices p U (h 2 ), p D (h 2 ).
To show how the shift from Q to Q * can be assessed quantitatively, rewrite both the numerator and denominator of the ratio of marginal utilities using first-order Taylor expansions about the mean (assuming higher-order terms can be ignored). Defining ( ) 2 m U D = + , the ratio of marginal utilities can thus be written is the Arrow-Pratt index of absolute risk aversion. Note that when risk aversion is introduced the reduction in p U (relative to a risk neutral equlibrium) cannot be greater than indicated by (A2.5). On the other hand, (A2.5) decreases as the Arrow-Pratt index increases, so the gap between Q and Q * increases as risk aversion increases. Theoretical Economics Letters Since changes in risk aversion affect Q * , and since earlier work has shown that changes in risk affect P, the model can distinguish the effects of the two types of change. Moreover, if the utility's risk aversion index can be determined independently of the model, it is also possible to determine whether a shift in the original function Q is caused by a change in expectations, by a change in risk preferences, or by a combination of the two.
To illustrate the effect of a change in risk preferences in the economy on upstate claim purchases, we show below that ( ) ( ) Since 0 λ > and 0 J > , the coefficient of the bracketed term is positive. Note that if investors are constant absolute risk averse (CARA), the expression is zero.
That is, changes in CARA investor risk preferences, even if economy-wide, do not affect their consumption decisions.
A more realistic assumption, however, is that investors are decreasing absolute risk averse (DARA). Then, we have d d 0 , which indicates that both speculators and hedgers decrease their portfolio diversification as the level of r decreases. If the level of r continues to decrease, investor portfolios become increasingly specialized, approaching those of risk neutral investors.
To illustrate the effects of an increase in market risk on upstate claim demand, change the investor decision problem of this section by adding a mean-preserving spread, as follows: