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.

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 investors believe they can profit from what they view as the irrationality of others. As already mentioned, in our model investors accept an equilibrium ruling price as given, and in that context the HK complications do not apply.

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.

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] examine value and cross-sectional momentum strategies across a wide variety of instruments and markets, finding 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 an information event. However, since momentum pricing eventually reverses, it assumes further that momentum investors return to rational investing in equilibrium.

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, U > D > 0 . There is a continuum of investors h ∈ ( 0 , 1 ) . 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 h ∈ [ 0 , 1 ] is assumed to have subjective probabilities ( q U ( h ) , q D ( h ) ) such that

q U ( h ) + q D ( h ) = 1 , (1)

for the realized state at t = 1. We further assume that q U ( h ) is a monotonically and strictly increasing function of h; hence q D ( h ) = 1 − q U ( h ) is strictly monotonically decreasing.⁴

To depict the impacts of changing expectations as simply as possible, we employ linear functions q U ( h ) and q D ( 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

∫ 0 1 C U h d h = U

∫ 0 1 C D h d h = D . (2)

We assume in attaining equilibrium, that all investors are risk-neutral and that each maximizes an expected utility:

U h ( C U , C D ) = q U ( h ) C U + q D ( h ) C D . (3)

Thus claims are valued at time zero using the equilibrium prices p U and p D 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 h 1 is defined as the marginal buyer who is indifferent between buying up or down claims at the prices p U and p D . Using (1) and (2), it is easy to verify that investors h ∈ [ h 1 , 1 ) will buy only upstate claims, and investors h ∈ ( 0 , h 1 ) only downstate claims. Consistent with prior literature, we call the former investors speculators, the latter hedgers.⁵ The speculators’ cash constraint⁶ is

∫ h 1 1 ( U p U + D p D ) d h = ( U p U + D p D ) ( 1 − h 1 ) = U p U

while the hedgers’ is

∫ 0 h 1 ( U p U + D p D ) d h = ( U p U + D p D ) ( h 1 ) = D p D (4)

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 is defined by investor proportions ( h 1 , 1 − h 1 ) . Since the marginal buyer h 1 is indifferent to buying the up or the down claim, the equilibrium must satisfy

q U ( h 1 ) p U ( h 1 ) = q D ( h 1 ) p D ( h 1 ) . (5)

It is convenient to rewrite (5) as

p U ( h 1 ) p D ( h 1 ) = q U ( h 1 ) q D ( h 1 ) , (6)

thus distinguishing cash constraint from expectations effects. To interpret the cash constraint side of (6), note that for any h 1 ∈ ( 0 , 1 ) both sides of (3) and (4) are positive quantities. Hence we can take the ratio of (3) to (4) to write

P ( h ) ≡ p U ( h ) p D ( h ) = D ( 1 − h ) U h (7)

with P(h) defining the price impact of the cash constraints for any investor h ∈ ( 0 , 1 ) . Note that P(h) increases with U. That is, holding probabilities constant, fewer high-state claims increase their price.

Note also from (7) that since P ( 1 / 2 ) = D / U < 1 , a necessary condition for an equilibrium with P > 1 is

( 1 − h ) > h ,

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

Q ( h ) ≡ q U ( h ) q D ( h ) . (8)

Then treating D and U as fixed, equating the expressions in (7) and (8) permits finding the equilibrium value h₁ , from which P(h₁), p_U(h₁), p_D(h₁), Q(h₁), q_U(h₁), and q_D(h₁) can all be determined. Note that changing expectations can only affect equilibrium prices and quantities if the marginal investor is affected. The risk premia are a consequence of the equilibrium condition Q ( h 1 ) = P ( h 1 ) .

Henceforth, we denote the equilibrium claim price ratio using P e 1 ≡ P ( h 1 ) . After a parameter change a new equilibrium will be denoted by P_e2. Similarly, we denote the original equilibrium ratio of speculators to hedgers by H e 1 ≡ ( 1 − h 1 ) / h 1 ; and after a parameter change by H_e2.

For any h ∈ ( 0 , 1 ) , (7) can be rewritten

( 1 − h ) h = U p U ( h ) D p D ( h ) (9)

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 = ( D p D U p U + D p D ) ;     1 − h = ( U p U U p U + D p D ) ; (10)

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 p U > 0 , p D > 0 , p U + p D = 1 ,

D < D p D + U p U < U ; (11)

the total value of securities expenditures increases with p_U . In (9), (10), and (11) equilibrium values are obtained by setting h = h 1 .

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.

Rewriting (8) as

Q ( h ) = q U ( h ) q D ( h ) = 1 − q D ( h ) q D ( h ) = 1 q D ( h ) − 1 .

and recalling that q ′ D ( h ) < 0 establishes Q ′ ( h ) > 0 ; Q ″ ( h ) > 0 , as drawn in Figure 1. Similarly, rewriting (7) as

P ( h ) = ( D U ) ( 1 h − 1 )

and treating U and D as parameters establishes P ′ ( h ) < 0 , P ″ ( h ) > 0 . Although 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.

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.

If the proportion of speculators making different adjustments remains the same as before, any rotation of Q is about h₁. In this case the marginal investor and claim prices would not change, but investors with different strategies could still react differently. For example, we would expect that investors following a momentum strategy would have different prior probabilities than those following a value strategy, contributing to different investment decisions. To the extent that an information event increases the upstate probability of all investors q_u(h) will shift up, and vice versa. In subsequent analyses we omit this possibility and concentrate on rotations attributable to ambiguous information events.

Rotation of the expectations function will not normally be around the existing marginal investor. If the rotation is counter clockwise and greater than h₁, a greater proportion of investors will raise their upstate probabilities. The probability of the new marginal investor h 2 < h 1 ; the proportion of speculators increases with the information being interpreted as favorable news by most investors. Alternatively, if the rotation is counter clockwise and h 3 > h 1 , the proportion of investors lowering their upstate probabilities increases relative to the original position and the information event is interpreted as bad news by most investors. In the rest of paper we shall study increasing agreement, in which case all rotations will be clockwise.

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.

Figure 2 indicates an initial equilibrium for which Q 2 ( h 2 ) < Q 1 ( h 1 ) , as the result of an information event⁷ regarding which investor agreement increases. This means that Q₂ slopes less steeply than Q₁, rotating clockwise around an intersection between Q₁(h) and a horizontal, homogeneous expectations line (for clarity the latter is not shown). Under these assumptions, the new Q₂(h) determines an equilibrium price ratio P_e2 such that P e 1 < P e 2 < P * , where P * refers to the homogeneous expectations price. The proportion of speculators also increases, in an amount dependent on the elasticity of Q₂(h). Depending on initial choices of data, the increased upstate claim purchases may be associated with either value or momentum criteria, according to how individual investors react to the information event. In a series of such adjustment processes, successive rotations reduce the slope in a series of steps that eventually bring Q_t closer to the horizontal (homogeneous expectations) line.

What sort of information event could trigger the behavior in Figure 2? An ambiguous information release (e.g., a central bank discussion of monetary policy) could cause speculators to lower their upstate probabilities and hedgers to increase theirs. As a result, investor agreement increases. If the resulting locus of

rotation is as shown in Figure 2, the upstate probability for the marginal investor decreases, the marginal investor moves to the left in the Figure, the number of speculators increases, and the greater speculative demand increases upstate claim prices.

Now consider a price decrease. In this case, suppose that after an information event occurs, Q(h) rotates clockwise with a locus of rotation h 2 < h 1 as shown by the intersection in Figure 3. The equilibrium price ratio decreases, P e 1 > P e 2 > P * , and speculative purchases also decrease. As before, these changes could be associated either with sale of an overvalued asset as represented by a reversal of contrarian purchases or by sales following on declining momentum criteria.

To further interpret Figure 3, suppose a different information event, one that is interpreted as bad news by most speculators and hedgers (e.g., a forecast of bad news regarding real sector performance). At the same time, a minority takes a contrarian view, believing that the forecast is an attempt to lower market expectations. Again, investor agreement increases, but the new locus of rotation is at a lower point h 2 < h 1 , the new marginal investor increases and the proportion of speculators decreases as do upstate claim prices. Complementing Miller’s (1977) argument that prices of riskier securities might change in either direction as opinion diverges, we have suggested that the ambiguity of an information event—signalled by a decrease in the slope of q_u(h)—will be a determinant of market behavior. Furthermore, a specific information event could lead to either an increase, a decrease, or no change in price.

We can also use Figure 3 to analyze the finding of CGP [2] that the price impacts

of the 2000-2002 tech crash were due principally to changes in discount rates rather than in expectations. A major contributing factor to the crash was a collapse of investor confidence due to numerous financial reporting failures (e.g., Enron, WorldCom, Nortel), all of which contributed to increased perceptions of market risk (investors bear market risk in our risk neutral economy unless claim prices are equal). Booming markets leading up to the crash created an increasing proportion of speculators, particularly in shares of (presumed high payoff) high tech companies. An increasing discount rate would lower speculators’ expectations closer to those of hedgers, driving a clockwise rotation of Q(h). The result is a reduction in upstate prices and a lower proportion of speculators.

CGP [2] also report that the price impacts in early stages of the 2007-2009 crash were due principally to changes in fundamentals (cash flow forecasts). As noted in Section 2.3 our model can also analyze this finding, but with a different approach. Arguably, boom times leading up to 2007 resulted in a large proportion of speculators. As market failures mounted in 2007, expected cash flows for all firms fell, with those of high payoff firms particularly affected. From (7), holding expectations constant, the result is to shift P(h) upward. This shift of P(h) leads to a lower proportion of speculators, with resulting fall in the share prices of former high payoff firms.