انتخاب ثروت محور در یک بازار مالی با عوامل نامتجانس
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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Journal of Economic Behavior & Organization, Volume 73, Issue 3, March 2010, Pages 327–358
We study the co-evolution of asset prices and individual wealth in a financial market with an arbitrary number of heterogeneous boundedly rational investors. Using wealth dynamics as a selection device we are able to characterize the long run market outcomes, i.e., asset returns and wealth distributions, for a general class of competing investment behaviors. Our investigation illustrates that market interaction and wealth dynamics pose certain limits on the outcome of agents’ interactions even within the “wilderness of bounded rationality”. As an application we consider the case of heterogeneous mean-variance optimizers and provide insights into the results of the simulation model introduced by Levy, Levy and Solomon (1994).
Consider a financial market where a group of heterogeneous investors, each following a different strategy to gain superior returns, is trading. The open questions are to specify how their interaction affects market returns and who will survive in the long run. This paper seeks to give a contribution to this issue by investigating the co-evolution of asset prices and agents’ wealth in a stylized market for a long-lived financial asset with an arbitrary number of heterogeneous agents. We do so under three main assumptions. First, asset demands are proportional to agents’ wealth, so that market clearing prices and agents’ wealth co-evolve. Second, each individual investment behavior can be formalized as a function of past returns. Third, the dividends of the risky asset follow a geometric random walk. By focusing on asset prices dynamics in a market with heterogeneous investors, our paper clearly belongs to the growing field of Heterogeneous Agent Models (HAMs), see Hommes (2006) for a recent survey. We share the standard set-up of this literature and assume that agents decide whether to invest in a risk-free bond or in a risky financial asset.2 In the spirit of Brock and Hommes (1997) and Grandmont (1998) we consider a stochastic dynamical system and analyze the sequence of temporary equilibria of its deterministic skeleton. Whereas the majority of HAMs consider only a few types of investors and concentrate on heterogeneity in expectations, our framework can be applied to a quite large set of investment strategies so that heterogeneity with respect to risk attitude, expectations, memory and optimization task can be accommodated. Employing the tools developed in Anufriev and Bottazzi (2009) we are able to characterize the long-run behavior of asset prices and agents’ wealth for a general set of competing investment strategies, which can be specified as a function of past returns. An important feature of our model concerns the demand specification. In contrast to many HAMs (see, e.g., Brock and Hommes, 1998, Gaunersdorfer, 2000 and Brock et al., 2005), which employ the setting where agents’ demand exhibits constant absolute risk aversion (CARA), we assume that demand increases linearly with agents’ wealth; that is, it exhibits constant relative risk aversion (CRRA). In such a setting agents affect market price proportionally to their relative wealth. As a consequence, relative wealth represents a natural measure of performance of different investment behaviors. On the contrary, in CARA models the wealth dynamics does not affect agents’ demand, implying that a performance measure has to be introduced ad hoc time by time. Furthermore, experimental literature seems to lean in favor of CRRA rather than CARA (see, e.g., Kroll et al., 1988 and Chapter 3 in Levy et al., 2000). The analytical exploration of the CRRA framework with heterogeneous agents is difficult because the wealth dynamics of every agent has to be taken into account. Although there has been some progress in the literature (see, e.g., Chiarella and He, 2001, Chiarella and He, 2002, Anufriev et al., 2006 and Anufriev, 2008 and Anufriev and Bottazzi, 2009), all these studies are based on the assumption that the price-dividend ratio is exogenous. This seems at odd with the standard approach, where the dividend process is exogenously set, while the asset prices are endogenously determined. In our paper, to overcome this problem, we analyze a market for a financial asset whose dividend process is exogenous, so that the price-dividend ratio is a dynamic variable. Our paper can thus be seen as an extension of Anufriev and Bottazzi (2009) to the case of exogenous dividends.3 As a result we show that depending on the difference between the growth rate of dividends and the risk-free rate, which are the exogenous parameters of our model, the dynamics can converge to two types of equilibrium steady-states. When the growth rate of dividend is higher than the risk-free rate, the equilibrium dividend yield is positive, asset gives a higher expected return than the risk-free bond, and only one or few investors have a positive wealth share. Only such “survivors” affect the price in a given steady-state. However, multiple steady-states with different survivors and different levels of the dividend yield are possible, and the range of possibilities depends on the whole ecology of traders. Otherwise, when the dividends’ growth rate is smaller than the risk-free rate, the dividend yield goes to zero, both the risky asset and the risk-free bond give the same expected return, and the wealth of all agents grows at the same rate as asset prices. Notice, however, that convergence to either types of steady-state equilibria is not granted. We show how local stability of each steady-state depends on the strength of the price feedback, occurring via the investment functions. An important reason for departing from previous works with CRRA demands is that it allows for a direct application to a well known simulation model. In fact, our CRRA setup with exogenous dividend process is identical to the setup of one of the first agent-based simulation model of a financial market introduced by Levy, Levy and Solomon (LLS model, henceforth); see, e.g., Levy et al. (1994). Their work investigates whether stylized empirical findings in finance, such as excess volatility or long periods of asset overvaluation, can be explained by relaxing the assumption of a fully-informed, rational representative agent. Despite some success of the LLS model in reproducing the financial “stylized facts”, all its results are based on simulations. Our general setup can be applied to the specific demand schedules used in the LLS model and, thus, provides an analytical support to its simulations. As we are looking at agents’ survival in a financial market ecology, our work can be also classified within the realm of evolutionary finance. The seminal work of Blume and Easley (1992), as well as more recent papers of Sandroni, 2000, Hens and Schenk-Hoppé, 2005, Blume and Easley, 2006 and Blume and Easley, 2009 and Evstigneev et al. (2006), investigate how beliefs about the dividend process affect agents’ dominance in the market. A key difference between our model and the evolutionary finance approach is that our agents can condition their investment decisions on past values of endogenous variables such as prices. As a consequence, in our framework prices today influence prices tomorrow through their impact on agents’ demands, generating a price feedback mechanism. In the HAMs such mechanism plays an important role for the stability of dynamics. For instance, when the investment strategy is too responsive to price movements, fluctuations are typically amplified and unstable price dynamics are produced. Indeed, we show that local stability is related to how far agents look in the past. This paper is organized as follows. Section 2 presents the model and leads to the definition of the stochastic dynamical system where prices and wealths co-evolve. The steady-states of the deterministic version of the system are studied in Section 3 when only one or two investors are trading. At this level of the analysis the investment behavior is left unspecified, but the process of wealth accumulation enable us to characterize the locus of possible steady-states and specify conditions for their local stability. Section 4 applies the former analysis to the special case where agents are mean-variance optimizers. Section 5 extends the analysis of Section 3 to the general case of NN investors. Section 6 uses these last analytical results to explain the simulations of the LLS model. Section 7 summarizes our main findings and concludes. Most proofs are collected in Appendices at the end of the paper.
نتیجه گیری انگلیسی
In his recent survey, LeBaron (2006) stresses that agent-based models do not require analytical tractability (as opposed to Heterogeneous Agents Models) and, therefore, are more flexible and realistic for what concerns their assumptions. In this paper we show that flexibility can be achieved in an analytically-tractable heterogeneous agent framework too. In fact, we have performed an analytic investigation of a stylized model of a financial market where an arbitrary set of investors is trading. Under the assumption that the impact of different agents on the market depends on their wealth shares, we have derived existence and stability results for a general set of investment functions and an arbitrary number of agents. Due to the selecting role of wealth dynamics, we have been able to characterize the steady-state equilibria of this economy. They can either lead to different average return for the risky and the riskless assets, in which case they are at the intersection of each agent Equilibrium Investment Function with the so called Equilibrium Market Curve, or lead to the same average return for the two assets. We have also shown that our analysis of a deterministic market dynamics is helpful for the understanding of its simulations with a stochastic dividend process. Having assumed exogenous growth dividend process, we have reached two research objectives. First, we have been able to investigate which features of previous results using the same framework, such as Anufriev et al. (2006) or Anufriev and Bottazzi (2009), are due to their assumption of a constant dividend yield and which are of a more general nature. Second, we have provided an analytical support of the LLS simulations, which would have not be possible within the CRRA framework developed so far. As for the first objective, we have shown that the specification of the dividend process does play a role in shaping the price and dividend dynamics. In our framework, the prices grow at a rate that is derived from the exogenous parameters, while the ecology of trading behaviors can affect only the dividend yield. In previous CRRA contributions on the other hand, the dividend yield is fixed and the ecology of behaviors affects the growth rate of prices and, consequently, dividends. At the same time the common CRRA framework, with a wealth-driven selection and a coupled price-wealth dynamics, is responsible for similarities such as the existence of a low-dimensional locus of possible steady-states, i.e., Equilibrium Market Curve (though of a different shape). On the way to our second objective, we have considered an example with mean variance maximizers characterized by two parameters: degree of risk aversion and memory span used to estimate future returns. When the growth rate of dividends is larger than the risk-free rate, the agents with the lowest risk aversion dominate the market, provided that their memory spans are big enough. As a result the market dynamics converge to the stable steady-state equilibrium, where prices are growing as fast as the dividends, and lower the risk aversion the higher the value of the dividend yield. In this case price fluctuations are due to the exogenous fluctuations of dividends. Otherwise, when the memory is not long enough, agents with different investment strategies coexist and the price fluctuations are endogenously determined. When, instead, the growth rate of dividends is smaller than the risk-free rate, steady-state equilibrium asset returns are equal to risk-free returns and the dividend yield converges to zero, no matter the ecology of agents. As a result wealths returns are equal for all investment functions and there is no selection. We have also explained that these differences in the market dynamics and selecting regime are due to the exogenous inflow of wealth in the economy, both through dividends and riskless returns.