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A decision-theoretic model of asset-price underreaction and overreaction to dividend news

Abstract

We combine new developments in decision theory with a standard consumption-based asset-pricing framework. In our model the efficient market hypothesis is violated if and only if agents’ beliefs express ambiguity about the stochastic process driving economic fundamentals. Asset price fluctuations result because agents with ambiguous beliefs are prone to a confirmatory bias in the interpretation of new information. We demonstrate that our approach gives rise to price-patterns of “underreaction” and “overreaction” to news about dividend payments. Although these empirical phenomena have received significant attention in the behavioral finance literature, we argue that our decision-theoretic underpinning of psychological attitudes has a less ad hoc flavor than existing approaches.

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Notes

  1. For empirical evidence on underreaction see, e.g., Bernard and Thomas (1990), Cutler et al. (1991), Bernard (1992), Jegadeesh and Titman (1993), Rouwenhorst (1997) , Chan et al. (1997) and Chan (2003). For empirical evidence on overreaction compare, e.g., De Bondt and Thaler (1985), Zarowin (1989), Chopra et al. (1992), La Porta (1996), La Porta et al. (1997) and Chateauneuf et al. (2007), Antweiler and Frank (2006).

  2. Also note that Wu and Gonzalez (1999) report Ellsberg-type behavior (more specifically: inversely \(S\)-shaped decision weighting functions) for bets on the future value of the Dow Jones IndustrialAverage.

  3. As explained in Sect. 2, this “indeterminacy” of update rules is a direct consequence of the violation of Savage’s sure thing principle (Savage 1954) as elicited in paradoxes of the Ellsberg (1961) type.

  4. In Sect. 2 we give precise definitions of these principles and derive their formal relationship to Savage’s sure thing principle which is at the heart of (subjective) EU theory.

  5. Observe that an CEU decision maker of our model who resolves his ambiguity exclusively in a pessimistic way (i.e., via a convex neo-additive capacity) could be equivalently described within the Gilboa and Schmeidler (1989) multiple priors framework if the priors are given as the core of this neo-additive capacity. However, in contrast to the original Gilboa and Schmeidler (1989) multiple priors framework—

    which can only express ambiguity aversion—our approach is in line with empirical evidence that suggests a mix between ambiguity proneness and ambiguity aversion [cf. the inversely \(S\)-shaped weighting functions as described by Wu and Gonzalez (1996, 1999) and by Wakker (2004)].

  6. We are grateful to an anonymous referee who pointed us to this literature.

  7. Intuitively speaking, null events are the events that the decision maker deems impossible. For instance, within a Savage framework we have that \(A\in \mathcal N \) if and only if, for all Savage acts \(f,g,h,h^{\prime }\),

    $$\begin{aligned} h_{A}f\succeq h_{A}^{\prime }g\Leftrightarrow f\succeq g, \end{aligned}$$

    i.e., consequences on \(A\) are irrelevant to the decision maker; (see Sect. 2.2 for the notation).

  8. This finding restates an implication of Lemma 2 in Ghirardato (2002) for our slightly different definitions of consequentialism and dynamic consistency.

  9. For an update rule for non-additive probability measures that violates consequentialism but satisfies dynamic consistency see Sarin and Wakker (1998b).

  10. As convergence of \(\prod _{s=0}^{t}\pi (y_{s})\rightarrow 0\) is fast, this would be reached in few periods only, even for very low levels of initial ambiguity, e.g., \(\delta _{0}=1.0E^{-10}\).

  11. Rabin (1998) reports empirical studies which do not support the conjecture that learning necessarily decreases biases. The models of biased Bayesian learning developed in Zimper and Ludwig (2009), Zimper (2009); Zimper (2011) describe an agent’s posterior neo-additive estimator for the likelihood of an event as the conditional Choquet expected value of the (unique) likelihood random variable taking on values in \(\left[ 0,1\right] \). For alternative learning models with Bayesian overtones that may result in biased (i.e., ambiguous) long-term estimators see Epstein and Schneider (2007, 2008) who introduce multiple likelihoods and Epstein et al. (2010) who consider behaviorally motivated ad hoc deviations from rational Bayesian learning.

  12. In a different application, Ludwig and Zimper (2012) relax such a convergence result by postulating limited memory. Using an equivalent assumption here would imply that prices in the optimistic, respectively pessimistic, regime would be determined by some weighted average between the maximal, respectively minimal, asset value and the rational expectations outcome. Qualitatively our results would, however, not change.

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Acknowledgments

We thank Martin Barbie for helpful discussions. We are also grateful to the insightful comments of an anonymous referee. Financial support from ERSA (Economic Research Southern Africa) is gratefully acknowledged.

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Correspondence to Alexander Ludwig.

Appendix

Appendix

Deriving the equilibrium price process

Observe that—independent of the specific choice of \(\left( z_{t},z_{t+1},{\ldots }\right) \)—the maximal, respectively minimal, economic outcome occurs for every portfolio-choice in the same state \(\left( \max Y_{1},\max Y_{2},{\ldots }\right) \in \varOmega \) resp. \(\left( \min Y_{1},\min Y_{2},{\ldots }\right) \in \varOmega \). Because of this fact and because of our assumption that every agent \(I_{t}\) assumes that \(\nu ^{I_{t}}\left( \cdot \mid \cdot \right) \) governs the beliefs of his future selves, we can transform the neo-additive Choquet expected utility maximization problem into an equivalent (standard) expected utility maximization problem for conveniently constructed additive probability measures \(\pi _{opt}^{I_{t}}\), \(\pi _{pess}^{I_{t}}\) and \(\pi _{FB}^{I_{t}}\). We thereby obtain the following equivalent maximization problems for the respective pricing regimes.

Optimistic pricing regime:

$$\begin{aligned}&\max _{\left( z_{t},z_{t+1},{\ldots }\right) }u\left( c_{t}\right) +E\left[\, \sum _{s=t+1}^{\infty }\beta ^{s-t}u\left( c_{s}\right) ,\nu ^{I_{t}}\left( \cdot \mid \cdot \right) \right] \\&\quad =\max _{\left( z_{t},z_{t+1},{\ldots }\right) }u\left( c_{t}\right) +E\left[\, \sum _{s=t+1}^{\infty }\beta ^{s-t}u\left( c_{s}\right) ,\pi _{opt}^{I_{t}}\left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \right] \end{aligned}$$

such that

$$\begin{aligned}&\pi _{opt}^{I_{t}}\left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \nonumber \\&\quad =\left\{ \begin{array}{ll} \delta _{opt}^{I_{t}}+\left( 1-\delta _{opt}^{I_{t}}\right) \cdot \pi \left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right)&\quad \text{ for}\quad y_{s+1}=\max Y_{s+1} \\ \left( 1-\delta _{opt}^{I_{t}}\right) \cdot \pi \left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right)&\quad \text{ for}\quad y_{s+1}<\max Y_{s+1}\\ \end{array} \right. \end{aligned}$$

Pessimistic pricing regime:

$$\begin{aligned}&\max _{\left( z_{t},z_{t+1},{\ldots }\right) }u\left( c_{t}\right) +E\left[\, \sum _{s=t+1}^{\infty }\beta ^{s-t}u\left( c_{s}\right) ,\nu ^{I_{t}}\left( \cdot \mid \cdot \right) \right] \\&\quad =\max _{\left( z_{t},z_{t+1},{\ldots }\right) }u\left( c_{t}\right) +E\left[\, \sum _{s=t+1}^{\infty }\beta ^{s-t}u\left( c_{s}\right) ,\pi _{pess}^{I_{t}}\left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \right] \end{aligned}$$

such that

$$\begin{aligned}&\pi _{pess}^{I_{t}}\left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \nonumber \\&\quad =\left\{ \begin{array}{ll} \delta _{pess}^{I_{t}}+\left( 1-\delta _{pess}^{I_{t}}\right) \cdot \pi \left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right)&\quad \text{ if}\quad y_{s+1}=\min Y_{s+1} \\ \left( 1-\delta _{pess}^{I_{t}}\right) \cdot \pi \left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right)&\quad \text{ if}\quad y_{s+1}>\min Y_{s+1}.\\ \end{array} \right. \end{aligned}$$

Full Bayesian pricing regime:

$$\begin{aligned}&\max _{\left( z_{t},z_{t+1},{\ldots }\right) }u\left( c_{t}\right) +E\left[\, \sum _{s=t+1}^{\infty }\beta ^{s-t}u\left( c_{s}\right) ,\nu ^{I_{t}}\left( \cdot \mid \cdot \right) \right] \\&\quad =\max _{\left( z_{t},z_{t+1},{\ldots }\right) }u\left( c_{t}\right) +E\left[\, \sum _{s=t+1}^{\infty }\beta ^{s-t}u\left( c_{s}\right) ,\pi _{FB}^{I_{t}}\left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \right] \end{aligned}$$

such that

$$\begin{aligned}&\pi _{FB}^{I_{t}}\left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \\&\quad =\left\{ \begin{array}{ll} \delta _{FB}^{I_{t}}\cdot \lambda +\left( 1-\delta _{FB}^{I_{t}}\right) \cdot \pi \left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right)&\quad \text{ if}\quad y_{s+1}=\max Y_{s+1} \\ \left( 1-\delta _{FB}^{I_{t}}\right) \cdot \pi \left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right)&\quad \text{ if}\quad \min Y_{s+1}<y_{s+1}<\max Y_{s+1} \\ \delta _{FB}^{I_{t}}\cdot \left( 1-\lambda \right) +\left( 1-\delta _{FB}^{I_{t}}\right) \cdot \pi \left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right)&\quad \text{ if}\quad y_{s+1}=\min Y_{s+1}.\\ \end{array} \right. \end{aligned}$$

Given this transformation of the CEU into an EU maximization problem the above results follow readily from standard arguments. For sake of completeness we demonstrate this for the optimistically biased pricing regime. Consider an optimistically biased agent so that

$$\begin{aligned}&E\left[\, \sum _{s=t+1}^{\infty }\beta ^{s-t}u\left( c_{s}\right) ,\nu ^{I_{t}}\left( \cdot \mid \cdot \right) \right] \\&\quad =\delta _{opt}^{I_{t}}\cdot \max _{\omega \in I_{t}}\sum _{s=t+1}^{\infty }\beta ^{s-t}u\left( c_{s}\right) +\left( 1-\delta _{opt}^{I_{t}}\right) \cdot \nonumber \\&\quad E\left[\, \sum _{s=t+1}^{\infty }\beta ^{s-t}u\left( c_{s}\right) ,\pi \left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \right]. \end{aligned}$$

The corresponding period \(s\) first order conditions, evaluated at equilibrium allocation \(z_{t}^{*}=1\) for all \(t\), imply

$$\begin{aligned} p_{s}^{*}&= \delta _{opt}^{I_{t}}\cdot \beta \cdot \frac{u^{\prime }\left( \max Y_{s+1}\right) }{u^{\prime }\left( y_{s}\right) }\cdot \left( \max Y_{s+1}+p_{s+1}^{*}\right) \\&+\left( 1-\delta _{opt}^{I_{t}}\right) \cdot E\left[ \beta \cdot \frac{ u^{\prime }\left( Y_{s+1}\right) }{u^{\prime }\left( y_{s}\right) }\cdot \left( Y_{s+1}+p_{s+1}^{*}\right) ,\pi \left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \right] \\&= \delta _{opt}^{I_{t}}\cdot \min M_{s,s+1}\cdot \left( \max Y_{s+1}+p_{s+1}^{*}\right) \\&+\left( 1-\delta _{opt}^{I_{t}}\right) \cdot E\left[ M_{s,s+1}\cdot \left( Y_{s+1}+p_{s+1}^{*}\right) ,\pi \left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \right] \end{aligned}$$

for all \(s\ge t\). Notice that

$$\begin{aligned} \min M_{s,s+1}=\beta \cdot \frac{u^{\prime }\left( \max Y_{s+1}\right) }{ u^{\prime }\left( y_{s}\right) } \end{aligned}$$

follows from concavity of \(u\). As a consequence, any period \(t\) equilibrium asset price \(p_{t}^{opt}\) is characterized by the following system of equations

$$\begin{aligned} p_{s}^{opt}&= E\left[ M_{s,s+1}\cdot \left( Y_{s+1}+p_{s+1}^{opt}\right) ,\pi _{opt}^{I_{t}}\left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \right] \\&\equiv\quad E_{s}^{opt}\left[ M_{s,s+1}\cdot \left( Y_{s+1}+p_{s+1}^{opt}\right) \right] \end{aligned}$$

for all \(s\ge t\) with \(\pi _{opt}^{I_{t}}\left( y_{s+1}\mid y_{0},\ldots ,y_{s}\right) \) defined above. Substitute

$$\begin{aligned} p_{t+1}^{opt}=E_{t+1}^{opt}\left[ m_{t+1,t+2}\cdot \left( Y_{t+2}+p_{t+2}^{opt}\right) \right] \end{aligned}$$

in

$$\begin{aligned} p_{t}^{opt}=E_{t}^{opt}\left[ M_{t,t+1}\cdot \left( Y_{t+1}+p_{t+1}^{opt}\right) \right] \end{aligned}$$

and observe that

$$\begin{aligned} p_{t}^{opt}&= E_{t}^{opt}\left[ M_{t,t+1}\cdot \left( Y_{t+1}+E_{t+1}^{opt} \left[ M_{t+1,t+2}\cdot \left( Y_{t+2}+p_{t+2}^{opt}\right) \right] \right) \right] \\&= E_{t}^{opt}\left[ M_{t,t+1}\cdot Y_{t+1}\right] +E_{t}^{opt}\left[ M_{t,t+1}\cdot E_{t+1}^{opt}\left[ M_{t+1,t+2}\cdot \left( Y_{t+2}+p_{t+2}^{opt}\right) \right] \right] \\&= E_{t}^{opt}\left[ M_{t,t+1}\cdot Y_{t+1}\right] +E_{t}^{opt}\left[ E_{t+1}^{opt}\left[ M_{t,t+1}\cdot M_{t+1,t+2}\cdot \left( Y_{t+2}+p_{t+2}^{opt}\right) \right] \right] \\&= E_{t}^{opt}\left[ M_{t,t+1}\cdot Y_{t+1}\right] +E_{t}^{opt}\left[ E_{t+1}^{opt}\left[ M_{t,t+2}\cdot \left( Y_{t+2}+p_{t+2}^{opt}\right) \right] \right] \\&= E_{t}^{opt}\left[ M_{t,t+1}\cdot Y_{t+1}\right] +E_{t}^{opt}\left[ M_{t,t+2}\cdot Y_{t+2}\right] +E_{t}^{opt}\left[ M_{t,t+2}\cdot p_{t+2}^{opt} \right]. \end{aligned}$$

The third line results from the fact that the random variable \(M_{t,t+1}\) is a constant with respect to any given \(E_{t+1}^{opt}\) and the fifth line is implied by the law of iterative expectations for additive probability measures. Applying the same reasoning to \(p_{t+2}^{opt},p_{t+2}^{opt},{\ldots }\) gives the desired result whenever the transversality condition is satisfied.\(\square \)

Remark

The key to the formal derivation of the equilibrium prices in the above proof is the fact that our assumptions allow us to transform the Choquet expected utility optimization problems into equivalent expected utility optimization problems. Hence, standard arguments such as sufficient characterization of global optima by first order conditions as well as the law of iterated expectations go through. This formal equivalence would break down if we had considered a portfolio choice problem with several assets that do not have a comonotonic payoff-structure. Technically speaking, the corresponding portfolio optimization problem would then exhibit kinks so that first order conditions are no longer sufficient criteria for global optima.

Formal proof of the proposition

  • Step 1. Observe that

    $$\begin{aligned}&E\left[ Y_{t+1},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right]\nonumber \\&\quad =E\left[ Y_{t+1},\pi \left( \cdot \mid y_{t-j-1}<y_{t-1}>{\cdots }>y_{t}\right) \right] \\&\quad =E\left[ Y_{t+1},\pi \right] , \end{aligned}$$

    for \(j\ge 1\), by the independence assumption.

  • Step 2. Fix some \(n\) and consider some \(j<n\). Then \( y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\) implies \(p_{t}^{*}=p_{t}^{FB}\) so that

    $$\begin{aligned}&E\left[ \frac{p_{t+1}^{*}+Y_{t+1}}{p_{t}^{*}},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right] \\&\quad =\frac{E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right] +E\left[ Y_{t+1},\pi \right] }{ p_{t}^{FB}}. \end{aligned}$$

    Similarly, \(y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\) implies

    $$\begin{aligned}&E\left[ \frac{p_{t+1}^{*}+Y_{t+1}}{p_{t}^{*}},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right] \\&\quad =\frac{E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right] +E\left[ Y_{t+1},\pi \right] }{ p_{t}^{FB}}. \end{aligned}$$

    Consequently,

    $$\begin{aligned}&E\left[ R_{t+1},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right]\nonumber \\&>E\left[ R_{t+1},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right] \end{aligned}$$

    is satisfied, if and only if,

    $$\begin{aligned}&E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right] \nonumber \\&>E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right]. \end{aligned}$$
    (34)
  • Step 3. Consider the case \(j<n-1\). Then \(p_{t+1}^{*}=p_{t+1}^{FB}\) for good as well as for bad news so that

    $$\begin{aligned}&E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right]\nonumber \\&\quad =E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right] \\&\Leftrightarrow&\\&E\left[ R_{t+1},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right] \nonumber \\&\quad =E\left[ R_{t+1},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right]. \end{aligned}$$

    Consequently, there is neither underreaction nor overreaction. This proves (26).

  • Step 4. Consider the case \(j=n-1\). Then \(p_{t+1}^{*}\) results for good news in period \(t\) either from the optimistic or from the full Bayesian pricing regime, i.e.,

    $$\begin{aligned}&E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right] \\&\quad =E\left[ p_{t+1}^{opt},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}<y_{t+1}\right) \right] \cdot \pi \left( y_{t}<y_{t+1}\right) \\&\quad +\,E\left[ p_{t+1}^{FB},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\ge y_{t+1}\right) \right] \cdot \pi \left( y_{t}\ge y_{t+1}\right) \end{aligned}$$

    For bad news in period \(t\), \(p_{t+1}^{*}\) follows either from the pessimistic or from the full Bayesian pricing regime, i.e.,

    $$\begin{aligned}&E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right] \\&\quad =E\left[ p_{t+1}^{pess},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}>y_{t+1}\right) \right] \cdot \pi \left( y_{t}>y_{t+1}\right) \\&\qquad +\,E\left[ p_{t+1}^{FB},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\le y_{t+1}\right) \right] \cdot \pi \left( y_{t}\le y_{t+1}\right). \end{aligned}$$

    If the histories satisfy condition (27), then

    $$\begin{aligned} \pi \left( y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}<y_{t+1t}\right)&> 0\,\text{ or} \\ \pi \left( y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}>y_{t+1}\right)&> 0. \end{aligned}$$

    Together with (23), the lhs in (34) first-order stochastically dominates the lhs in (34) so that we obtain underreaction for \(j=n-1\). If, instead, (28) holds, then

    $$\begin{aligned} \pi \left( y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}<y_{t+1}\right)&= 0\,\text{ and} \\ \pi \left( y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}>y_{t+1}\right)&= 0. \end{aligned}$$

    This proves (29).\(\square \)

  • Step 5. Consider the case \(j\ge n\). Then \(p_{t}^{*}=p_{t}^{opt} \) for history \(y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\) and \(p_{t}^{*}=p_{t}^{pess}\) for history \(y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\). Consequently,

    $$\begin{aligned}&E\left[ R_{t+1},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right] \\&\quad =\frac{E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right] +E\left[ Y_{t+1},\pi \right] }{ p_{t}^{opt}} \end{aligned}$$

    and

    $$\begin{aligned}&E\left[ R_{t+1},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right] \\&\quad =\frac{E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right] +E\left[ Y_{t+1},\pi \right] }{ p_{t}^{pess}}. \end{aligned}$$

    Because

    $$\begin{aligned} \frac{E\left[ Y_{t+1},\pi \right] }{p_{t}^{opt}}<\frac{E\left[ Y_{t+1},\pi \right] }{p_{t}^{pess}}, \end{aligned}$$

    our model generates overreaction (3) if the sufficient condition

    $$\begin{aligned}&\frac{E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right] }{p_{t}^{opt}}\nonumber \\&\quad \quad \le \frac{E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right] }{p_{t}^{pess}} \end{aligned}$$
    (35)

    is satisfied. If \(j\ge n\), \(p_{t+1}^{*}\) results for good news in period \(t\) either from the optimistic or full Bayesian pricing regime, i.e.,

    $$\begin{aligned}&E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\right) \right] \\&\quad =E\left[ p_{t+1}^{opt},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}<y_{t+1}\right) \right] \cdot \pi \left( y_{t}<y_{t+1}\right) \\&\qquad +E\left[ p_{t+1}^{FB},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}\ge y_{t+1}\right) \right] \cdot \pi \left( y_{t}\ge y_{t+1}\right). \end{aligned}$$

    Accordingly, \(p_{t+j}^{*}\) results for bad news either from the pessimistic or full Bayesian pricing regime, i.e.,

    $$\begin{aligned}&E\left[ p_{t+1}^{*},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\right) \right] \\&\quad =E\left[ p_{t+1}^{pess},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}>y_{t+1}\right) \right] \cdot \pi \left( y_{t}>y_{t+1}\right) \\&\qquad +E\left[ p_{t+1}^{FB},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\le y_{t+1}\right) \right] \cdot \pi \left( y_{t}\le y_{t+1}\right). \end{aligned}$$

    Substituting in inequality (35) and rearranging gives

    $$\begin{aligned}&\left( \frac{E\left[ p_{t+1}^{opt},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}<y_{t+1}\right) \right] }{p_{t}^{opt}}\right.\nonumber \\&\qquad \left.-\frac{E \left[ p_{t+1}^{FB},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}\le y_{t+1}\right) \right] }{p_{t}^{pess}}\right) \nonumber \\&\qquad \cdot \pi \left( y_{t}<y_{t+1}\right) \nonumber \\&\quad \le \left( \frac{E\left[ p_{t+1}^{pess},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}>y_{t+1}\right) \right] }{p_{t}^{pess}}\right.\nonumber \\&\qquad \left.-\frac{E \left[ p_{t+1}^{FB},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}>y_{t+1}\right) \right] }{p_{t}^{opt}}\right) \nonumber \\&\qquad \cdot \pi \left( y_{t}>y_{t+1}\right) \nonumber \\&\qquad +\,\left( \frac{E\left[ p_{t+1}^{FB},\pi \left( \cdot \mid y_{t-j-1}<y_{t-j}>{\cdots }>y_{t}=y_{t+1}\right) \right] }{p_{t}^{pess}}\right.\nonumber \\&\qquad \left.-\frac{E \left[ p_{t+j}^{FB},\pi \left( \cdot \mid y_{t-j-1}>y_{t-j}<{\cdots }<y_{t}=y_{t+j}\right) \right] }{p_{t}^{opt}}\right) \nonumber \\&\qquad \cdot \pi \left( y_{t}=y_{t+1}\right) \end{aligned}$$
    (36)

    By the i.i.d. assumption, \(\lim _{t\rightarrow \infty }\pi \left( I_{t}\right) =0\) for all \(I_{t}\), which implies

    $$\begin{aligned} \lim _{t\rightarrow \infty }\delta _{opt}^{I_{t}}=\lim _{t\rightarrow \infty }\delta _{pess}^{I_{t}}=\lim _{t\rightarrow \infty }\delta _{FB}^{I_{t}}=1 . \end{aligned}$$

    As a consequence, the respective equilibrium prices of any given regime converge to some constant, i.e.,

    $$\begin{aligned} \lim _{t\rightarrow \infty }p_{t}^{opt}&= \frac{\beta }{1-\beta }\cdot \max Y , \\ \lim _{t\rightarrow \infty }p_{t}^{pess}&= \frac{\beta }{1-\beta }\cdot \min Y, \\ \lim _{t\rightarrow \infty }p_{t}^{FB}&= \frac{\beta }{1-\beta }\cdot \left( \lambda \cdot \max Y+\left( 1-\lambda \right) \cdot \min Y\right). \end{aligned}$$

    By continuity, the expected prices in (36) satisfy

    $$\begin{aligned}&E\left[ p_{t+1}^{opt},\pi \left( \cdot \mid \cdot \right) \right]\simeq p_{t}^{opt}\simeq \frac{\beta }{1-\beta }\cdot \max Y, \\&E\left[ p_{t+1}^{FB},\pi \left( \cdot \mid \cdot \right) \right] \simeq p_{t}^{FB}\simeq \frac{\beta }{1-\beta }\cdot \left( \lambda \cdot \max Y+\left( 1-\lambda \right) \cdot \min Y\right) \\&E\left[ p_{t+1}^{pess},\pi \left( \cdot \mid \cdot \right) \right] \simeq p_{t}^{pess}\simeq \frac{\beta }{1-\beta }\cdot \min Y, \end{aligned}$$

    for sufficiently large \(t\). Substituting in (36) gives, for sufficiently large \(t\),

    $$\begin{aligned}&\left( 1-\frac{p_{t}^{FB}}{p_{t}^{pess}}\right) \cdot \pi \left( y_{t}<y_{t+1}\right) \le \left( 1-\frac{p_{t}^{FB}}{p_{t}^{opt}}\right) \cdot \pi \left( y_{t}>y_{t+1}\right) \\&\quad +\left( \frac{p_{t}^{FB}}{p_{t}^{pess}}-\frac{p_{t}^{FB}}{p_{t}^{opt}} \right) \cdot \pi \left( y_{t}=y_{t+1}\right) , \end{aligned}$$

which is always satisfied with strict inequality since the lhs is negative and the rhs is positive because of \(p_{t}^{opt}>p_{t}^{FB}>p_{t}^{pess}\). This proves overreaction for \(j\ge n\) if \(t\) is sufficiently large. \(\square \)

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Ludwig, A., Zimper, A. A decision-theoretic model of asset-price underreaction and overreaction to dividend news. Ann Finance 9, 625–665 (2013). https://doi.org/10.1007/s10436-012-0208-z

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Keywords

  • Choquet expected utility theory
  • Portfolio choice
  • Asset pricing puzzles
  • Overreaction
  • Underreaction

JEL Classification

  • C62
  • D81
  • G11
  • G12