Markovian process

In the case that we called uninformed we assumed that the realization of the state the world at a time t depends on the realization observed previously, at a time t − 1. In particular we studied analytically the case of symmetric transition probabilities: that is, observing a certain realization, for example assume it is positive, if δ is positive it is more likely that the subsequent realization will be positive as well, on the other hand, it is more likely to revert if δ is negative.We start the experiment simulating the time series of wealth, prices and related absolute return in the case of positive δ, in particular we set δ = 0.3. Looking at Figure 3.11, which shows the autocorrelagram and the partial autocorrelogram, it is evident that wealth distribution is positively autocorrelated, confirming our analytical findings.

However, as for the informed case, the main aim is to examine the dynamics of the absolute return. In the case of δ > 0 we the autocorrelogram of returns shown in Figure 3.13 confirm our expectations, returns are positively correlated in very short-run implying that the sign of return could be a good predictor for the subsequent return, but in the short-run tend to be negatively autocorrelated. Such results are consistent with several empirical studies (Cutler, Poterba, and Summers 1991), which found that such positive autocorrelation exsists only in the very short run.

Let now turn to the case of δ < 0. In particular we set δ = −0.3 and then we run the simulation. Both the wealth distribution and the price ditribution follow the pattern we derived in the analytical study (Section 3.4.2). In fact, as shown in Figure 3.15,wealth distributions is positively autocorrelated. For what concern the evolution of returns series, Figure 3.17 shows that returns are negatively autocorre-lated in the short-run and then such autocorrelation tend to disappear over longer horizons. This result confirms our analytical findings, and it is also consistent with a mean-reverting behavior of returns treated in the literature (Poterba and Summers (1988), Lakonishok, Shleifer, and Vishny (1994)). As a consequence we have shown that in a representative agent framework in which the evolution of the states of the

3.5 Simulation Experiments and Implications 77

world follows a Markov process, the risk-aversion of the agent that operates in the market, can cause the emergence of signals of inefficiency in the stock market.

78 A Representative Agent Model

Figure 3.10: Wealth time-series for the last 100 periods, x = 0.5, λ = 0.5, δ = 0.3

Figure 3.11: Wealth autocorrelogram lag = 10, δ = 0.3

3.5 Simulation Experiments and Implications 79

Figure 3.12: Returns series for the last 100 periods, x = 0.5, λ = 0.5, δ = 0.3

Figure 3.13: Returns autocorrelogram lag = 10, δ = 0.3

80 A Representative Agent Model

Figure 3.14: Wealth series for the last 100 periods, x = 0.8, λ = 0.5, δ = −0.3

Figure 3.15: Wealth autocorrelogram lag = 10, δ = −0.3

3.5 Simulation Experiments and Implications 81

Figure 3.16: Returns series for the last 100 periods, x = 0.5, λ = 0.5, δ = −0.3

Figure 3.17: Returns autocorrelogram lag = 10, δ = −0.3

Conclusion

he last 30 years have been very busy for academic finance. Financial markets have been studied under several points of view and now we have a knowledge of their behavior and of market operators that is nearly complete. Among the many changes of views, the increased skepticism about market efficiency has been a central issue although it has been very controversial. Such skepticism derives from many sources that have put the basis to a new discipline: Behavioral Finance.

Behavioral finance has provided both theory and evidence which suggest what deviations of securities from fundamental values are likely to be, and why the can persist over long time without being eliminated. In many cases, behavioral theo-ries have been developed on the basis of the strong and relevant empirical findings, discussed in Chapter 1, challenging the efficient market hypothesis. Although empir-ical evidence was the real first instrument of behavioralists against efficient market theorists, the debates over market efficiency continue, so that behavioral finance derive its strength by its behavioral explanations of the so-called anomalies, which materialized in theories of investor behavior. The study of the psychology of market participants has been absolutely innovative, that has provided theories and expla-nation in many cases incontestable. The usefulness of behavioral finance lies in offering a richer description of investor behavior than those captured by fully ratio-nal utility maximizers with limited heterogeneityfor example in risk preferences by giving a collection of possible heuristics and biases, which have been documented in a financial , or sometimes a more general, decision-making setting. The relevance of each behavioral phenomenon should be treated as a research question on its own and addressed using appropriate techniques.

84 Conclusion

In this context take place several behavioral models that have been developed during the last 20 years, which try to explain, on the theoretical ground, the emer-gence of market anomalies, return predictability, underreaction and overreaction, momentum strategy and others. This thesis aims to contribute to this literature, providing a detailed representative agent model. Despite many behavioral models, as those discussed in Chapter 2, our model does not take into account particu-lar biases that can affect the investors behaviors, such as representativeness and conservatism in Barberis, Shleifer, and Vishny (1998) or overconfidence and self-attribution in Daniel, Hirshleifer, and Subrahmanyam (1998), rather it wonders if and to what extent, a risk-averse agent can be considered has biased in its beliefs and then determine stock price deviations or returns autocorrelation. In our model there is a representative risk-averse agent, which behaves as expected utility maxi-mizers and invests a constant fraction of wealth in a short-lived risky asset. Further we assumed that the world can be in two states, so that we studied two different cases: one in which the two states are equiprobable and independent from previous realization and one in which the realization of the states depend only on the previous realization. After deriving analytically the dynamics of wealth, price and returns, we simulated the model. The results of the simulation showed that in both cases returns are autocorrelated, implying that returns are not completely unpredictable as stated by market efficiency, even when agent has not biased beliefs.

To enhance the implications of our findings the hope of the candidate is to goes ahead in its academic career, learning and acquire more instrument and techniques to extend such model, for example introducing an agent allowing then for heterogeneity, and adding more risky assets in order to allow a portfolio diversification.

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