Survival in speculative markets

https://doi.org/10.1016/j.jet.2019.02.002Get rights and content

Abstract

In this paper, I consider an exchange economy with complete markets where agents have heterogeneous beliefs and, possibly, preferences, and investigate the Market Selection Hypothesis that speculation rewards the agent with the most accurate beliefs. First, on the methodological level, I derive the relative consumption dynamics as a function of agents' effective discount factors, related to consumption decisions across time, and agents' effective beliefs, related to consumption decisions across states. Sufficient conditions for agents' survival, either in isolation or in a group, depend on the relative size of effective discount factors and on the relative accuracy of effective beliefs. Then, I show that in economies where agents maximize an Epstein–Zin utility the Market Selection Hypothesis fails: there exist parametrizations where the agent with correct beliefs vanishes and parametrizations where beliefs heterogeneity persists in the long run. Results are robust to local changes of beliefs, risk preferences, and the aggregate endowment process. These failures are shown not to occur when agents' Epstein–Zin utility has a subjective expected utility representation due to an interdependence of effective discount factors and effective beliefs.

Introduction

The dominant academic view of financial markets is that they facilitate hedging and risk diversification. A complementary view is that trade also occurs due to investors' disagreement about the return distribution of assets. Indeed, also in standard models of financial markets such as Lucas' model or the CAPM, agents' beliefs heterogeneity makes them willing to hold risky positions different from those they would have held under pure hedging. These positions are speculative, in that they include a bet on future realizations of assets' fundamentals.

Although speculative motives are certainly present, a widespread position of financial economists is that speculation can be only a transient characteristic of markets. The Market Selection Hypothesis (MSH) of Friedman (1953) applied to financial markets presumes that investors with accurate beliefs can earn high returns by taking positions against investors with inaccurate beliefs. These speculative positions should thus allow accurate traders to dominate and bring asset prices at the fundamental value implied by their beliefs. Indeed, when markets are complete, each agent can freely trade to allocate her future consumption on the paths which she believes more likely. In equilibrium, everything else being equal, the agent with the most accurate beliefs assigns the highest likelihood to paths that are actually realized and, thus, should hold everything in the long run. In bounded economies with time-separable Subjective Expected Utility (SEU) maximizers, the argument is rigorously established by Sandroni (2000) and Blume and Easley (2006). However, despite the importance of the result, the exact nature of the role of risk and time preferences in its validity is still unclear. Moreover, a number of contributions show that outside the SEU framework with bounded aggregate endowment the MSH may fail.2

In this paper, I investigate the effect of speculation on agents' relative consumption dynamics by identifying the separate roles of optimal consumption decisions across time and states. The contribution is twofold. Firstly, on the methodological side, I derive the relative consumption dynamics as a function of agents' effective discount factors, related to consumption decisions across time, and agents' effective beliefs, related to consumption decisions across states. Sufficient conditions for agents' survival, either in isolation or in a group, depend on the relative size of a survival index that takes into account the size of the effective discount factor and the accuracy of effective beliefs. As the intuition above suggests, the agents with the most correct beliefs are still those who, assigning the highest likelihood to paths that are actually realized, do hold everything in the long run. However, the beliefs that matter are the effective ones, and those are endogenous, in that, depending on risk preferences, they also incorporate equilibrium prices.

Secondly, and related to the MSH, I show that when, for at least one agent, consumption decisions across time and states are not interdependent, e.g. when there is at least an agent who maximizes a recursive Epstein–Zin utility, the MSH fails: there exist economies where agents with heterogeneous beliefs survive in the long run, or others where the agent with correct beliefs vanishes on a set of paths with full measure, even when risk and time preferences are homogeneous. Path dependency, with an agent dominating on some paths and another on others is also a possible long-run outcome. Results are robust to local changes of beliefs, risk preferences, and the aggregate endowment process. These failures are shown not to occur when the Epstein–Zin utility has a subjective expected utility representation, due to the interdependence of consumption decisions across time (effective discount factors) and states (effective beliefs).

After having introduced the general set-up (Section 2), the methodological contribution is developed in Section 3 where I focus on 2-agent economies. The full analysis of I-agent economies proceeds along the same lines but, being slightly heavier on the formal side, is postponed to Section 5.

In Section 3, I show that the dynamics of consumption shares depends on two key quantities: the ratio between the current value of next period consumption and current consumption, which accounts for optimality across time, and the ratio between the current value of next period consumption in each state and the current value of next period total consumption, which accounts for optimality across states. For an SEU maximizer with log Bernoulli utility, the first ratio (time dimension) is equal to her discount factor while the second ratio (states dimension) is equal to her beliefs. More generally, these ratios, which can be derived from the Euler equations, depend also on equilibrium state prices and time/risk preferences. Yet, they can be interpreted as an effective discount factor (for the time dimension) and effective beliefs (for the states dimension). In fact, they lead to the same consumption decision of an SEU-log agent who uses them, respectively, as discount factor and beliefs.

Sufficient conditions for an agent to survive, dominate, or vanish can be written in terms of the log of effective discount factors and accuracy of effective beliefs. I name the sum of these two terms the generalized survival index. The size of the effective discount factor matters for survival because it is related to saving and depends, other than on the discount factor, on the intertemporal elasticity of substitution (IES) and on the perceived mispricing. A unitary IES implies that perceived mispricing has no effect on saving, since income and substitution effect compensate each other. Larger (smaller) IES encourages (depresses) saving in the presence of perceived mispricing. The larger the mispricing, the larger the effect.

The accuracy of effective beliefs matters as well because it is related to the agent portfolio expected log-returns and is shown to depend both on the accuracy of beliefs and on an endogenous component that I name Non-Log-Optimality (NLO) term. Unless for agents with unitary relative risk aversion (RRA), who hold log-optimal portfolios, the NLO term is endogenous (it depends, through risk preferences, on state prices) and measures the relative accuracy of effective beliefs and beliefs. An agent with correct beliefs and non-unitary RRA has always a negative NLO term, because effective beliefs are less accurate than beliefs. An agent with non-correct beliefs may have instead a positive (negative) NLO term when beliefs and preferences imply effective beliefs that are closer (further away) to the truth than beliefs. A typical case of a positive NLO term is when optimism and risk aversion compensate each-other. Failures of the MSH depend on the interplay of this NLO contribution with the belief dependent component of the effective discount factor.

Examples of failure of the MSH are presented in Section 4, where I consider specific parametrizations of Epstein–Zin recursive utility. The examples provided are aimed to shed light on the separate role of effective discount factors (decision across time, saving) and effective beliefs (decision across states, portfolios) about MSH failures.

A parametrization which is particularly useful to clarify why the MSH can fail has all agents with unitary IES and equal discount factors, so that effective discount factors are homogeneous, but with different effective beliefs. In economies where all agents have RRA higher than 1 and there is aggregate risk, optimism may compensate risk aversion and the agent with non-correct beliefs may dominate due a high NLO term (Section 4.1). However, if optimism is too strong the outcome is long-run heterogeneity because each agent has a high NLO term, and also more accurate effective beliefs, at the state prices determined by the other agent. I provide examples of long-run heterogeneity in 2-agent economies in Section 4.2. The intuition for 2-agent economies works also for I-agent economies as shown in the examples of Appendix A.3.

Path dependency occurs instead when both agents have an RRA lower than 1 and the non correct agent is a pessimist, because each agent typically has a negative NLO term, and also less accurate effective beliefs, at the prices determined by the other agent. I provide examples of path dependency in 2-agent economies in Section 4.3.

Finally, in Section 4.4, I consider examples where all agents maximize an SEU-CRRA utility. As established by the incumbent literature, provided preferences are homogeneous, the agent with correct beliefs dominates. My contribution is to show that the result holds because, the IES coefficient being the inverse of the RRA coefficient, the endogeneity of effective beliefs (portfolios) and of effective discount factors (saving) compensate each other and what matters is only the accuracy of beliefs and the size of discount factors. In these economies, it is enough to slightly change the preferences of one agent, so that her RRA is not equal to the inverse of her IES, to obtain MSH failures such as long-run heterogeneity.

Section 5 extends the methodological contributions to I-agent economies. Appendix A presents further examples showing how failures are robust to local changes of beliefs, risk preferences, and the aggregate endowment process. Appendices B–D collect the proofs. In Appendix E, I compute effective discount factors and beliefs for general time-separable utility and, possibly, a non-linear probability weighting function. In the next section, I discuss the relation between my results and the incumbent literature.

Investors speculate when they take long and/or short positions that they would not have otherwise taken had they agreed on the underlying state process.3 A number of contributions investigate the short-term effect of speculation on asset prices and the volume of trade (see e.g Varian, 1985, Varian, 1989; Harris and Raviv, 1993; Kandel and Pearson, 1995) or the relation between speculation and financial innovations (see e.g. Zapatero, 1998; Brock et al., 2009; Simsek, 2013). Here, I am instead interested in whether speculation can also have long-run consequences. My conclusion is that, also in the idealized framework of general equilibrium with complete markets, unless all agents' optimal decisions across time and states are interdependent, speculation may be a persistent feature of financial markets. The result could help to explain stock market anomalies as recently suggested in Anderson et al. (2005), Hong and Stein (2007), Cogley and Sargent (2009), Yu (2011), Bhamra and Uppal (2014), Hong and Sraer (2016), Baker et al. (2016), and Bottazzi et al. (2018b).

The relation between speculation and the MSH for financial economies has received increasing attention since the works of DeLong et al., 1990, DeLong et al., 1991 and Blume and Easley (1992). DeLong et al. (1991) find that noise traders, i.e. traders with inaccurate beliefs, might survive or even dominate against rational traders by bearing more risk. The analysis is based on a partial equilibrium model. Blume and Easley (1992) study the same issue in a temporary equilibrium model and find that, controlling for the saving rate, when the trader with the most accurate beliefs purchases a log-optimal portfolio, she gains all the wealth in the long run and brings asset prices to reflect her beliefs. However, when the trader with the most accurate beliefs does not use the log-optimal rule, Blume and Easley are able to derive conditions for this trader to vanish.4

Subsequent work by Sandroni (2000) and by Blume and Easley (2006) extend the analysis to general equilibrium models with endogenous saving. Under the assumptions of markets' completeness, bounded aggregate endowment, and SEU maximization with time-separable payoff, the MSH holds: provided that all agents discount future utility at the same rate, only the trader with the most accurate beliefs dominates. Results are established by comparing a survival index that depends solely on the size of discount factors and beliefs accuracy.

A related contribution is Yan (2008), where the MSH is investigated in a continuous-time economy where the aggregate endowment follows a Brownian motion and agents have CRRA preferences. Agents agree on the volatility of the aggregate endowment process but disagree on its drift. The findings by Sandroni (2000) and Blume and Easley (2006) are confirmed: provided preferences are homogeneous the accurate trader dominates.5,6

By finding failures of the MSH even in a general equilibrium framework, my results provide rational foundations for the findings of the earlier studies by DeLong et al. (1991) and Blume and Easley (1992). Moreover, with respect to the general equilibrium literature, I show how the use of effective discount factors and effective beliefs allows the Euler equations to be rewritten as consumption dynamics, somehow similarly to the original approach in Blume and Easley (1992) for agents' wealth dynamics. Effective discount factors and effective beliefs, which can be defined also for groups of agents, disentangle the role of consumption decisions across time and states toward survival.

Other influential works investigate the MSH in economies without intermediate consumption; see in particular Kogan et al., 2006, Kogan et al., 2017 and Cvitanić and Malamud, 2010, Cvitanić and Malamud, 2011. Results in the two types of economies – with or without intermediate consumption – are known to differ due to the effect of saving in the former. Having clarified that decisions across time are not only important per se, but also for the fact that they might or might not compensate the endogenous NLO term of effective beliefs accuracy, my work reconciles the findings of the two literatures regarding the role of decisions across states.

In analyzing the MSH in Epstein–Zin economies, this paper is also related to Borovička (2019), who investigates the MSH in continuous-time exchange economies with two agents having homogeneous Epstein–Zin preferences. My results, in particular that MSH failures are possible and robust to changes in beliefs and risk preferences, confirm his findings. On the methodological side the two papers are, however, quite different. Borovička solves directly the central planner problem and characterizes agents' (general equilibrium) optimal policies exploiting the partial equilibrium limit of one agent being alone in the economy.7 I study long-run survival in economies with possibly more than two agents and provide sufficient conditions that can be used beyond the Epstein–Zin utility case. The generalized survival index and its decomposition in terms of the size of the effective discount factor and accuracy of effective beliefs holds in general.

Another related contribution is Easley and Yang (2015) where long-run outcomes are computed in a 2-agent economy with an Epstein–Zin investor and a loss-averse investor. Consistently with my results, it is shown that the loss-averse investor vanishes because her portfolio is further away from the log-optimal one. Moreover, the loss-averse agent can survive and dominate only when she saves more than the Epstein–Zin investor.

Within the market selection literature, other studies find long-run beliefs heterogeneity. Beker and Chattopadhyay (2010) and Cogley et al. (2013) focus on 2-agent economies with incomplete markets. Beker and Espino (2011) highlights the importance of learning. Cao (2018) studies an economy where markets are endogenously incomplete due to portfolio and collateral constraints. Guerdjikova and Sciubba (2015) study economies where investors are ambiguity averse. Bottazzi and Dindo (2014) and Bottazzi et al. (2018a) extend the temporary equilibrium analysis of Blume and Easley (1992) to general asset structures, short-lived and long-lived respectively, and possibly incomplete markets. Dindo and Massari (2017) present a model with collective learning where long-run heterogeneity emerges because agents use equilibrium prices to update their beliefs, making them endogenous.

Section snippets

The economy

In this section, I introduce a rather general model of an exchange economy with complete markets and heterogeneous agents and provide the definition of survival, dominance, and vanishing.

Time begins at date t=0 and is indexed by tN0={0,1,2,}. S={1,2,,S} is the set of states of the world that can occur at each date t>0 and stS denotes the state realized in t. Σ=×t=0S is the set of paths σ=(s1,s2.) and σt=(s0,s1,,st)Σt is the history till period t (a node in the tree representation). C(σt)

Relative consumption dynamics

Our objective is to study the relative consumption process and to provide conditions, as well as an underlying intuition, for an agent to survive, dominate, or vanish. In this section, I concentrate on 2-agent economies and prepare the ground for the key examples of MSH failure in Section 4. The analysis of I-agent economies is postponed to Section 5.

I start by characterizing an agent's consumption process in terms of her effective discount factor, related to consumption decisions across time,

Survival in Epstein–Zin economies

We shall apply the sufficient conditions established in the previous section to certain parametrization of economies with 2 agents having Epstein–Zin preferences. We have two objectives. First, we show that outside the SEU framework there exists a variety of long-run outcomes: dominance of the trader with non-correct beliefs (even when discount factors and inter-temporal preferences are homogeneous), persistent heterogeneity, and path dependency. Notably, all these outcomes are possible even if

Groups' consumption dynamics

In this section, we generalize the analysis to economies where I>2 and provide the sufficient conditions for survival that we have informally anticipated in Section 3.3 for the 2-agent economy. The analysis proceeds in parallel to that of Section 3 through the definition of group effective discount factors, effective beliefs, and generalized survival indexes.

Consider a subset GI with G traders and define, for all (t,σ), the group total consumption share as ΦtG=iGϕti and agent i relative

Conclusion

This paper explains why, in dynamic stochastic exchange economies where agents have heterogeneous beliefs, speculation may not support the Market Selection Hypothesis that the trader with the most accurate beliefs dominates in the long run.

The result is established by characterizing long-run outcomes of agents' relative consumption dynamics in terms of effective discount factors and effective beliefs. The accuracy of the latter is shown to determine the size of expected log-returns, or growth

References (56)

  • L. Kogan et al.

    Market selection

    J. Econ. Theory

    (2017)
  • F. Massari

    Trading in the market: a necessary and sufficient condition for a trader to vanish

    J. Econ. Dyn. Control

    (2017)
  • A. Sandroni

    Market selection when markets are incomplete

    J. Math. Econ.

    (2005)
  • J. Yu

    Disagreement and return predictability of stock portfolios

    J. Financ. Econ.

    (2011)
  • F. Zapatero

    Effects of financial innovations on market volatility when beliefs are heterogeneous

    J. Econ. Dyn. Control

    (1998)
  • E. Anderson et al.

    Do heterogeneous beliefs matter for asset pricing?

    Rev. Financ. Stud.

    (2005)
  • A. Araujo et al.

    On the convergence to homogeneous expectations when markets are complete

    Econometrica

    (1999)
  • H. Bhamra et al.

    Asset prices with heterogeneity in preferences and beliefs

    Rev. Financ. Stud.

    (2014)
  • L. Blume et al.

    If you are so smart why aren't you rich? Belief selection in complete and incomplete markets

    Econometrica

    (2006)
  • J. Borovička

    Survival and long-run dynamics with heterogeneous beliefs under recursive preferences

    J. Polit. Econ.

    (2019)
  • G. Bottazzi et al.

    Drift Criteria for Persistence of Discrete Stochastic Processes on the Line

    (2015)
  • G. Bottazzi et al.

    Long-run heterogeneity in an exchange economy with fixed-mix traders

    Econ. Theory

    (2018)
  • G. Bottazzi et al.

    Momentum and Reversal in Financial Markets with Persistent Heterogeneity

    (2018)
  • D. Cao

    Speculation and financial wealth distribution under belief heterogeneity

    Econ. J.

    (2018)
  • T. Cogley et al.

    Diverse beliefs, survival and the market price of risk

    Econ. J.

    (2009)
  • T. Cogley et al.

    Wealth dynamics in a bond economy with heterogeneous beliefs

    Econ. J.

    (2013)
  • J. Cvitanić et al.

    Financial market equilibrium with heterogeneous agents

    Rev. Finance

    (2012)
  • J. Cvitanić et al.

    Relative extinction of heterogeneous agents

    B.E. J. Theor. Econ.

    (2010)
  • Cited by (19)

    • Market selection and learning under model misspecification

      2023, Journal of Economic Dynamics and Control
    • Social contagion and the survival of diverse investment styles

      2023, Journal of Economic Dynamics and Control
    • Extrapolative asset pricing

      2023, Journal of Economic Theory
    • Self-enforcement, heterogeneous agents, and long-run survival

      2021, Economics Letters
      Citation Excerpt :

      Kogan et al. (2017) considers an economy in which there are two agents with the same utility function but have different priors, and analyzes the price impact of an agent as well as conditions under which an agent survives. Dindo (2019) shows that market selection hypothesis can fail in economies where agents have Epstein-Zin utilities. Massari (2019) shows that markets with a continuum of traders can favor luck traders with incorrect beliefs over skilled traders with accurate beliefs.

    • Heterogeneous beliefs with herding behaviors and asset pricing in two goods world

      2021, North American Journal of Economics and Finance
      Citation Excerpt :

      This argument is confirmed for expected utility by the work of Blume and Easley (2006) and Sandroni (2000). However, this argument does not generalize to recursive utility as shown in Borovicka (2020) and Dindo (2019) who show that the investors with wrong beliefs can also survive in the long run. Identically, we also show that the investors with wrong beliefs can survive in long run in presence of herding behaviors.

    View all citing articles on Scopus

    I wish to thank Pablo Beker, Larry Blume, Jaroslav Borovička, Giulio Bottazzi, David Easley, Daniele Giachini, Ani Guerdjikova, Filippo Massari, Jan Werner, and Alvaro Sandroni as well as Laura Veldkamp (the editor) and three anonymous referees for their helpful comments and suggestions. All errors are mine. I gratefully acknowledge the hospitality of the Department of Economics at Cornell University, where this project was developed. This research is supported by the Marie Curie International Outgoing Fellowship PIOF-GA-2011-300637.

    1

    Dipartimento di Economia, Università Ca' Foscari Venezia, Cannaregio 873, 30121 Venezia – Italy.

    View full text