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Momentum and reversal in financial markets with persistent heterogeneity

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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.

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Notes

  1. See Sect. 2 for an account of the empirical literature on momentum and reversal.

  2. See Sect. 2 for more details on empirical results concerning Winners–Losers portfolios.

  3. In doing so, we neglect the price impact of such portfolio strategies. Their inclusion in the set of trading strategies could be an interesting topic for future work.

  4. We shall show that this is the case in Sect. 4.1. We anticipate that having no aggregate risk is crucial for this result, otherwise returns should be risk adjusted.

  5. A similar aggregation is performed in the behavioral finance literature, see e.g. Barberis et al. (1998).

  6. Other works in the evolutionary finance literature assuming that states of nature follow a Markov process are Amir et al. (2005) and Evstigneev et al. (2006). These works do not investigate the emergence of momentum and reversal.

  7. The same analysis holds even when agents have a different discount factor. The only difference is that the risk-free rate depends on agents’ relative wealth. In the long-run, only the agent with the highest discount factor has positive wealth, and thus determines prices.

  8. Provided beliefs and discount factors are homogeneous, the exact functional form of U does not matter when the aggregate endowment is constant. Prices and portfolio shares can thus be derived using Euler equations and market clearing for any choice of the representative agent.

  9. As we shall characterize later, \((\mathcal {I}_t)\) can be restricted to \((\mathcal {F}_t)\) for our purposes, Conditional on such restriction, \(\mathbb {Q}\) can be derived from \(\mathbb {P}\). See the beginning of Sect. 5.

  10. In particular, since this learning process generically converges to the model with the lowest relative entropy with respect to the invariant \(\pi ^*\) (Berk 1966), we directly assume that each agent uses the best i.i.d. model from the beginning.

  11. The choice of i.i.d. models rather than, for example, Markov models is aimed at ensuring tractability. We try to be careful in understanding the effect of this choice on our results, especially on momentum, and will comment on it again in Sect. 5.1.

  12. As shown in Bottazzi et al. (2018), the rule becomes log-optimal under perfect foresight on prices when the agent using it becomes the representative agent. Under constant aggregate endowment, the “Subjective Generalized Kelly” is thus inter-temporally optimal for all risk-adverse Bernoulli utility \(U^i\), still provided perfect foresight on prices and a representative agent.

  13. Saving rule heterogeneity would give a survival advantage to the agent who has the highest saving rate.

  14. The definition of survival is in accordance with Sandroni (2000) and Bottazzi et al. (2018) among others. Depending on the model specifications, other different definitions of survival, dominance, and vanishing are possible, see e.g. Evstigneev et al. (2009, 2016).

  15. For a precise appraisal of the role of saving rates on agents’ portfolio positions we refer the reader to the discussion on effective beliefs in Bottazzi et al. (2018). Intuitively, simple portfolio rules can be mapped into log-optimal portfolio rules by modifying agents’ beliefs into effective endogenous beliefs. Higher values of saving rates imply a low interest rate and thus a high influence of assets’ future dividends for their evaluation. The same pair of simple rules need closer effective beliefs, and thus less severe speculation, in order to counterbalance the higher impact of future dividends.

  16. A detailed explanation of the procedure can be found in “Appendix E”.

  17. Sources of the data: http://www.multpl.com/s-p-500-dividend-yield/ for the dividend yield (whose yearly average is divided by 4 to get the quarterly one) and http://www.ck.dartmouth.edu/pages/faculty/ken.french/data_library.html for the 1-month risk free rate (which is compounded in order to find the 3-month one).

  18. Obviously excluding the zero-measure events in which \(p_t=1/2+\lambda \) and \(p_t=1/2-\lambda \) that yield a null expected return.

  19. Despite agent demand is not directly derived by an inter-temporal utility maximization, no arbitrages are possible in our model, see also Bottazzi et al. (2018).

  20. Different initial conditions do not affect the results.

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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).

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Correspondence to Pietro Dindo.

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Pietro Dindo has been supported by the Marie Curie International Outgoing Fellowship PIOF-GA-2011-300637 within the 7th European Community Framework Programme. Daniele Giachini gratefully acknowledges the hospitality of the Department of Banking and Finance at the University of Zurich. This research has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 640772—DOLFINS.

Appendices

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.

C Proof of Propositions 3

By Lemma 1 we can average expected returns with respect to the invariant distribution \(\pi ^*\) and this yields

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

which gives the required expression. The fact that \(\phi _t(1,p_t)>0\) for all \((t,\sigma )\) and for all \(p_t\in (\pi ^2,\pi ^1)\) implies \(\Phi _t(1)>0\) for all \((t,\sigma )\), irrespective of the price distribution. Subtracting Eq. (18) from Eq. (21) one gets

$$\begin{aligned} \frac{r_f\lambda (\pi ^1-p_t)(1-p_t+\delta (p_t-\pi ^2))}{\pi ^1(p_t(1-p_t)-\delta (\pi ^1-p_t)(p_t-\pi ^2))}\ge 0, \end{aligned}$$

with the equality holding if and only if \(\pi ^1=\pi ^2\). Thus, the statement follows.

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 conditionFootnote 20\(p_0=1/2\) 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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Bottazzi, G., Dindo, P. & Giachini, D. Momentum and reversal in financial markets with persistent heterogeneity. Ann Finance 15, 455–487 (2019). https://doi.org/10.1007/s10436-019-00353-0

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