Momentum and reversal in financial markets with persistent heterogeneity

Giulio Bottazzi; Pietro Dindo; Daniele Giachini

doi:10.1007/s10436-019-00353-0

Annals of Finance

December 2019, Volume 15, Issue 4, pp 455–487 | Cite as

Momentum and reversal in financial markets with persistent heterogeneity

Authors
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Giulio Bottazzi
Pietro Dindo
Daniele Giachini

Research Article

First Online: 08 October 2019

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Abstract

This paper investigates whether short-term momentum and long-term reversal may emerge from the wealth reallocation process taking place in speculative markets. We assume that there are two classes of investors who trade long-lived assets by holding constantly rebalanced portfolios based on their beliefs. Provided beliefs, and thus portfolios, are sufficiently diversified, all investors survive in the long-run and, due to waves of mispricing, the resulting equilibrium returns exhibit long-term reversal. If, moreover, asset dividends are positively correlated, investors’ profitable trades become positively correlated too, thus generating short-term momentum in equilibrium returns. We use the model to replicate the performance of the Winners and Losers portfolios highlighted by the empirical literature and to provide insights on how to improve upon them. Finally, we show that dividend positive autocorrelation is positively related to momentum and negatively related to reversal while diversity of beliefs is positively related to both momentum and reversal.

Keywords

Market efficiency Heterogeneous beliefs Speculation Short-term momentum Long-term reversal Evolutionary finance

JEL Classification

G60 D53 G02 G12 G14

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Notes

Funding

The funding was provided by Horizon 2020 Framework Programme (Grant No. 640772 - DOLFINS) and FP7 People: Marie-Curie Actions (Grant No. PIOF-GA-2011-300637).

A Proof of Proposition 1

Following the assumptions we made, one has

$$\begin{aligned} \mu (\pi ^a,\pi ^b,s)= & {} \left( \frac{1}{2}+(-1)^{\Delta _{2,s}}\lambda \right) \log \left( \frac{\delta \pi ^b+(1-\delta )\pi ^a}{\pi ^b}\right) \nonumber \\&\qquad \qquad \qquad \qquad +\left( \frac{1}{2}+(-1)^{\Delta _{1,s}}\lambda \right) \log \left( \frac{1-\delta \pi ^b-(1-\delta )\pi ^a}{1-\pi ^b}\right) .\nonumber \\ \end{aligned}$$

(22)

Since $0<\pi ^2<\pi ^1<1$, there exists a $\varepsilon >0$ such that $\varepsilon \le \pi ^2<\pi ^1\le 1-\varepsilon $. Let

$$\begin{aligned} z_t=\log \frac{w_t}{1-w_t}=\log \frac{p_t-\pi ^2}{\pi ^1-p_t} \; \end{aligned}$$

From (11), given that $p\in [\pi ^2,\pi ^1]$, it is easy to show that for $s_{t+1}=1$ it is

$$\begin{aligned} 0<z_{t+1}-z_t<\log \frac{1-\varepsilon }{\varepsilon } \end{aligned}$$

and for $s_{t+1}=2$ it is

$$\begin{aligned} \log \frac{\varepsilon }{1-\varepsilon }< z_{t+1}-z_t <0. \end{aligned}$$

Since $z_{t+1}-z_t$ is continuous in p, these inequalities imply that $(z_t)$ is a bounded increments process with finite positive and negative increments as defined in Bottazzi and Dindo (2015).

Call $\mathcal {M}(z,s)=E[z_{t+1}|z_{t}=z,s_t=s]-z$ and notice that $\lim _{z\rightarrow +\infty }\mathcal {M}(z,1)=-\mu (\pi ^2,\pi ^1,1)$, $\lim _{z\rightarrow +\infty }\mathcal {M}(z,2)=-\mu (\pi ^2,\pi ^1,2)$, $\lim _{z\rightarrow -\infty }\mathcal {M}(z,1)=\mu (\pi ^1,\pi ^2,1)$ and $\lim _{z\rightarrow -\infty }\mathcal {M}(z,2)=\mu (\pi ^1,\pi ^2,2)$. Thus, if the conditions stated in (15) are satisfied, Theorem 2.2 of Bottazzi and Dindo (2015) applies, the process $(z_t)$ is persistent and both agents survive. The statement in (16) follows from the finite positive and negative increments of $(z_t)$. Along the same lines, if the conditions in (17) are satisfied, either Theorem 3.1 or Corollary 3.1 of Bottazzi and Dindo (2015) apply, the process $(z_t)$ is transient and one of the two agents dominates while the other vanishes.

B Proof of Corollary 1

The conditions in (15) can be written as

$$\begin{aligned} \begin{array}{ll} \eta \left( \frac{1}{2}+\lambda ,\delta \pi ^b+(1-\delta )\pi ^a,\pi ^b\right) &{}>0,\\ &{}\\ \eta \left( \frac{1}{2}-\lambda ,\delta \pi ^b+(1-\delta )\pi ^a,\pi ^b\right) &{}>0, \end{array} \end{aligned}$$

with

$$\begin{aligned} \eta (\pi ,\pi ',\pi '')=\pi \log \frac{\pi '}{\pi ''}+(1-\pi )\log \frac{1-\pi '}{1-\pi ''} \;. \end{aligned}$$

By Jensen’s inequality, it is

$$\begin{aligned} \log \left( 1+\frac{(\pi ''-\pi ')(\pi ''-\pi )}{\pi ''(1-\pi '')}\right) > \eta (\pi ,\pi ',\pi '') \;. \end{aligned}$$

Thus, if the right hand side is positive, the argument of the logarithm in the left hand side must be greater than one. Substituting the values for $\pi $, $\pi '$ and $\pi ''$ and remembering that $\delta \in (0,1)$, this implies

$$\begin{aligned} (\pi ^b-\pi ^a)\left( \pi ^b-\frac{1}{2} -\lambda \right)>0 \quad \text {and} \quad (\pi ^b-\pi ^a)\left( \pi ^b-\frac{1}{2} +\lambda \right) >0 \end{aligned}$$

which, remembering that $0<\pi ^1<\pi ^2<1$, is equivalent to the assertion.

D Proof of Propositions 4

From Lemma 1, irrespective of the value of $p_t$, one has

$$\begin{aligned} \begin{array}{ll} \lim _{j\rightarrow \infty }E[r_{1,t+j}|\mathcal {R}^+_{t+j-1}(j),p_{t}]&{}=E[r_{1,t+1}|\mathcal {R}^+_{t}(1),\pi ^1]= \frac{1-\delta }{\delta }\;\frac{1/2+\lambda -\pi ^1}{\pi ^1},\\ \lim _{j\rightarrow \infty }E[r_{1,t+j}|\mathcal {R}^-_{t+j-1}(j),p_{t}]&{}=E[r_{1,t+1}|\mathcal {R}^-_{t}(1),\pi ^2]= \frac{1-\delta }{\delta }\;\frac{1/2-\lambda -\pi ^2}{\pi ^2}. \end{array} \end{aligned}$$

Averaging with respect the the invariant distribution of the states on the world $\pi ^*$ gives

$$\begin{aligned} \phi _\infty (p_t)= \frac{1}{2} E[r_{1,t+1}|\mathcal {R}^+_{t}(1),\pi ^1]- \frac{1}{2} E[r_{1,t+1}|\mathcal {R}^-_{t}(1),\pi ^2], \end{aligned}$$

which gives the required expression. Notice that this expression does not depend on $p_t$, hence $\Phi _\infty =\phi _\infty (p_t)$.

E Evaluation of winners and losers performances

We start by establishing a length J for the formation period, a length H for the performance period, and the total number of formation periods N. Then we numerically iterate the map in (11) for $T=J\times N+H$ periods starting with initial condition²⁰$p_0=1/2$ and compute for any time step $t=1,2,\ldots ,T$ the abnormal return of each asset k defined as

$$\begin{aligned} AR_{k,t}=r_{k,t}-\frac{r_{1,t}+r_{2,t}}{2}. \end{aligned}$$

For each formation period $n=1,\ldots ,N$, we compute the cumulative return of each asset k in that formation period as

$$\begin{aligned} CU_{k,n}=\sum \limits _{\tau =(n-1)J+1}^{nJ} AR_{k,\tau }. \end{aligned}$$

During the same time steps we rank the two assets on the basis of their cumulative return and assign them either to the Winners portfolio (W) or to the Losers portfolio (L). The Winners portfolio built after the n-th formation period buys asset 1 if $CU_{1,n}>CU_{2,n}$ and asset 2 if $CU_{2,n}>CU_{1,n}$. The Losers portfolio does the opposite. Then we evaluate the performance of the two portfolios during the following H periods computing the cumulative abnormal return for each $h=1,2,\ldots ,H$. Using the subscript w to indicate the winner asset and the subscript l to indicate the loser one, the cumulative abnormal returns for each time step h of the n-th performance period read

$$\begin{aligned} CAR^W_{n,h}=\sum \limits _{\tau =nJ+1}^{nJ+h} AR_{w,\tau },\quad CAR^L_{n,h}=\sum \limits _{\tau =nJ+1}^{nJ+h} AR_{l,\tau }. \end{aligned}$$

De Bondt and Thaler (1985), relying on a wide cross-section of assets, prescribes to build Winners and Losers portfolio considering only those assets that have extreme cumulative returns with respect to such cross-section. Having only two assets in our economy, we reproduce such feature using the time series dimension instead of the cross-section one. Knowing the time distribution of cumulative returns, we consider the performance of the two portfolios only after formation periods in which the cumulative return of one asset is below the first quartile (and thus it contributes to the loser portfolio) or above the third quartile (and thus it contributes to the winner portfolio). Naming M the set of formation periods for which the condition is respected, whose cardinality is N / 2 given the use of quartiles, we compute the average cumulative abnormal return of each portfolio

$$\begin{aligned} ACAR^W_{h}=\frac{\sum \limits _{n\in M} CAR^W_{n,h}}{N/2},\quad ACAR^L_{h}=\frac{\sum \limits _{n\in M} CAR^L_{n,h}}{N/2}, \end{aligned}$$

and the relative standard errors

$$\begin{aligned} SE^W_{h}=\sqrt{\frac{\sum \limits _{n\in M} \left( CAR^W_{n,h}-ACAR^W_{h}\right) ^2}{N/2(N/2-1)}},\quad SE^L_{h}=\sqrt{\frac{\sum \limits _{n\in M} \left( CAR^L_{n,h}-ACAR^L_{h}\right) ^2}{N/2(N/2-1)}}. \end{aligned}$$

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Giulio Bottazzi
- 1
Pietro Dindo
- 2
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Daniele Giachini
- 1

1.Dipartimento EMbeDS, Istituto di EconomiaScuola Superiore Sant’AnnaPisaItaly
2.Dipartimento di EconomiaUniversità Ca’ Foscari VeneziaVeneziaItaly

Momentum and reversal in financial markets with persistent heterogeneity

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