Price Dynamics in Market with Heterogeneous Investment Horizons and Boundedly Rational Traders

Transcription

1 Price Dynamics in Market with Heterogeneous Investment Horizons and Boundedy Rationa Traders Thierry Chauveau, Aexander Subbotin To cite this version: Thierry Chauveau, Aexander Subbotin. Price Dynamics in Market with Heterogeneous Investment Horizons and Boundedy Rationa Traders. Documents de travai du Centre d Economie de a Sorbonne ISSN : X. 21. <hashs > HAL Id: hashs Submitted on 5 Ju 21 HAL is a muti-discipinary open access archive for the deposit and dissemination of scientific research documents, whether they are pubished or not. The documents may come from teaching and research institutions in France or abroad, or from pubic or private research centers. L archive ouverte puridiscipinaire HAL, est destinée au dépôt et à a diffusion de documents scientifiques de niveau recherche, pubiés ou non, émanant des étabissements d enseignement et de recherche français ou étrangers, des aboratoires pubics ou privés.

2 Documents de Travai du Centre d Economie de a Sorbonne Price Dynamics in a Market with Heterogeneous Investment Horizons and Boundedy Rationa Traders Aexander SUBBOTIN, Thierry CHAUVEAU Maison des Sciences Économiques, bouevard de L'Hôpita, Paris Cedex 13 ISSN : X

3 Price Dynamics in a Market with Heterogeneous Investment Horizons and Boundedy Rationa Traders Aexander Subbotin Thierry Chauveau Apri 1, 21 Abstract This paper studies the effects of mutipe investment horizons and investors bounded rationaity on the price dynamics. We consider a pure exchange economy with one risky asset, popuated with agents maximizing CRRA-type expected utiity of weath over discrete investment periods. An investor s demand for the risky asset may depend on the historica returns, so that our mode encompasses a wide range of behaviorist patterns. The necessary conditions, under which the risky return can be a stationary iid process, are estabished. The compatibiity of these conditions with different types of demand functions in the heterogeneous agents framework are expored. We find that conditiona voatiity of returns cannot be constant in many generic situations, especiay if agents with different investment horizons operate on the market. In the atter case the return process can dispay conditiona heteroscedasticity, even if a investors are so-caed fundamentaists and their demand for the risky asset is subject to exogenous iid shocks. We show that the heterogeneity of investment horizons can be a possibe expanation of different styized patterns in stock returns, in particuar, mean-reversion and voatiity custering. Keywords: Asset Pricing, Heterogeneous Agents, Mutipe Investment Scaes, Voatiity Custering J.E.L. Cassification: G.1, G.14. Résumé Nous étudions effet de mutipes horizons d investissement et de a rationaité imitée des investisseurs sur a dynamique des prix. Dans e cadre d un modèe à agents hétérogènes, nous introduisons de mutipes échees d investissement. Nous trouvons que a voatiité conditionnee des rendements ne peut pas être constante dans de nombreuses situations génériques, en particuier si es agents avec des horizons d investissement différents opèrent sur e marché. Dans ce dernier cas, e processus de rendements peut être hétéroscédastique, même si tous es investisseurs sont fondamentaistes mais eur demande de actif risqué est soumise à des chocs iid exogènes. Nous montrons que hétérogénéité des horizons d investissement peut être une expication possibe de différents faits styisés, observés pour es rendements des actions et, en particuier, du retour à a moyenne et du custering de a voatiité. Mots cés: Evauation des actifs, agents hétérogènes, mutipes horizons d investissement, custering de voatiité. J.E.L. Cassification: G1, C13. The authors thank K. Shapovaova, M. Anufirev, C.Hommes, R.Topo, Y. Mirkin and B. de Meyer for hepfu remarks. University of Paris-1 (CES/CNRS), e-mai: aexander.subbotin@univ-paris1.fr University of Paris-1 (CES/CNRS), e-mai: thierry.chaveau@univ-paris1.fr 1 Documents de Travai du Centre d'economie de a Sorbonne

4 1 Introduction Up to now, the heterogeneous markets iterature amost excusivey focuses on the expectations of market agents, according to which investors are cassified into fundamentaists, chartists and noise traders. It is shown that the interaction, herding behavior and strategy switching of heterogeneous agents transform noise process and create persistent trading voume, excess voatiity, fat tais, custered voatiity, scaing aws (see Hommes [26] and LeBaron [26] for surveys on interacting agents modes). Andersen [1996] interprets the aggregated voatiity as the manifestation of numerous heterogeneous information arrivas. Limits to arbitrage, market psychoogy, heuristics and biases, which are subject of behaviora finance, can aso be hepfu to expain empirica evidence [see Barberis and Sheifer, 23]. A number of anayticay sovabe modes were proposed to expore the dynamics of financia market with heterogeneity coming from boundedy rationa beiefs of investors about future returns. Brock and Hommes [1998] proposed a mode, where investors switch between a number of strategies according to expected or reaized excess profits. Styized simpe strategies describe patterns in investors behavior that are commony observed empiricay - chartism and trend-foowing. Chiarea and He [21], Anufriev et a. [26] and Anufriev [28] studied an artificia market popuated with investors, foowing heterogeneous strategies and maximizing the expected CRRA utiity. Compared to earier studies that use CARA utiities, they make investment decisions depend on weath, which is undoubtedy more reaistic but technicay more difficut. Vanden [25] introduces a more sophisticated step-wise dependence of the risk aversion on weath and finds that this can have important consequences for return dynamics. Recenty Weinbaum [29] showed that heterogeneous risk preferences and risk sharing can be the source of voatiity custering. To our knowedge, a the above-mentioned modes of heterogeneity ignore one of its important sources, which is different investment scaes. By investment scaes we mean typica periods between two consecutive adjustments of investment portfoio, pecuiar to a certain type of investors. The heterogeneity of the market with respect to agents operations frequencies is further referred to as the Mutipe Investment Scaes (MIS) hypothesis. We suppose that investors maximize expected utiity of weath at the end of some investment period. We ca the typica ength of this period as investment horizon (or scae). Earier the effect of heterogeneity in investment horizons was studied in Anufriev and Bottazzi [24]. They derive a fixed point for the price of the risky asset dynamics under the assumption that agents maximize expected CARA utiity over different periods in future. But their mode disregards the effect of various frequencies of portfoio adjustments and, due to the constraints of the CARA assumption, does not reaisticay account for the dynamics of weath. They concude that heterogeneity of investment horizons aone is not enough to guarantee the instabiity of the fundamenta price and the emergence of the non-trivia price dynamics, such as voatiity custering or seria correations. In this paper we derive the opposite concusion, which is cose to that obtained in Chauveau and Topo [22]. Working in a different framework, they expained voatiity custering of OTC exchange rates by market microstructure effects, unifying intraday and interday dynamics. Though not examining the MIS hypothesis anayticay, severa earier stud- 2 Documents de Travai du Centre d'economie de a Sorbonne

5 ies evoke the heterogeneity of investment horizons as a possibe expanation of the styized facts in stock price voatiity. The assumption that price dynamics is driven by actions of investors at different horizons serves as a micro-economic foundation of the voatiity modes in Müer et a. [1997]. They suppose that there exist voatiity components, corresponding to particuar ranges of stock price fuctuation frequencies, that are of unequa importance to different market participants. These participants incude intraday specuators, daiy traders, portfoio managers and institutiona investors, each having a characteristic time of reaction to news and frequency of operations on the market. So frequencies of price fuctuations depend on the periods between asset aocation decisions, and/or the frequencies of portfoio readjustments by investors. An important question, answered in this paper, is whether the presence of (i) contrarian and trend-foowing investors and (ii) heterogeneous information arrivas on the market are necessary properties for an interacting agents mode to reproduce the styized facts of the return voatiity dynamics. We show that, under some conditions, voatiity custering can arise even in an economy popuated with fundamentaist traders ony, given that they adjust their portfoios with different frequencies. We aso propose a study of the joint effect of the MIS hypothesis and of the bounded rationaity in investment strategies. The rest of the paper is organized as foows. In the next section we introduce the genera setting of the mode. Section 3 describes the equiibria in the onescae mode with boundedy rationa investors, re-examining the concusions of Anufriev et a. [26] and preparing the ground for the study of the muti-scae case. In section 4 we derive the equiibrium in the MIS case and estabish the properties of the return dynamics. In section 5 we iustrate our findings with simuation exampes. In concusion the main resuts are summarized and possibe mode extensions are discussed. 2 A Mode for Joint Dynamics of Stock Price and Weath with Mutipe Investment Scaes In this section we formuate the mode and then discuss its various possibe specifications and assumptions. The genera setup foows the ines of Chiarea and He [21] and Anufriev et a. [26], to which we add the MIS hypothesis and some constraints on investors behavior, discussed ater. Where possibe, we keep the same notation as in Anufriev et a. [26], to enabe the easy comparison of resuts. Consider a two-assets market where N agents operate at discrete dates. The risk-free asset yieds a constant positive interest R f over each period and the risky asset pays dividend D t at the beginning of each period. The price of the risk-free asset is normaized to one and its suppy is absoutey eastic. The quantity of the risky asset is constant and normaized to one, whie its price is determined by market cearing by a Warasian mechanism. The Warasian assumption means that a agents determine their demand for the risky asset taking the price of the risky asset P t as parameter. In other, though this price is unobserved at the moment when investors form their demand, they cacuate the demand for the risky asset at every possibe price and submits this to a hypothetica Warasian auctioneer. The price is then set so that the tota demand 3 Documents de Travai du Centre d'economie de a Sorbonne

6 across a agents equas to one. The demand of the risky asset is formuated in terms of the shares of weath of agents, so that x t,i stands for the share of weath that investor i with weath W t,i wishes to invest in the risky asset. The corresponding number of units of the asset is Wt,ixt,i P t. The market cearing condition imposes: N x t,i W t,i = 1 i=1 The weath of each investor evoves according to the beow equation: W t,i = (1 x t 1,i )W t 1,i (1 + R f ) + x t 1,iW t 1,i P t 1 (P t + D t ) = (1 x t 1,i )W t 1,i (1 + R f ) + x t 1,i W t 1,i (1 + R t + ε t ), where D t is a dividend payment, whose ration to price is supposed to be an iid random variabe ε t, and R t is the return on the risky asset. We define the tota return by Y t = P t + D t P t 1. Foowing Anufriev et a. [26], we rewrite the mode in rescaed terms which aows to eiminate the exogenous expansion due to the risk-free asset growth from the mode: w t,i = W t,i (1 + R f ) t, p P t t = (1 + R f ) t, e t = ε t, y t = Y t. 1 + R f 1 + R f By consequence, the rescaed return on the risky asset is defined by: r t = p t p t 1 1 = 1 + R t 1 + R f 1 = R t R f 1 + R f. (1) In these terms the whoe system dynamics simpifies to: p t = i x t,i w t,i, w t,i =w t 1,i [1 + x t 1,i (r t + e t )]. (2) Proposition 2.1. The rescaed price dynamics, soving the dynamic system (2), verifies: i p t = p w t 1,i (x t,i x t 1,i x t,i ) + e t i x t,ix t 1,i w t 1,i t 1 i w, t 1,i (x t 1,i x t,i x t 1,i ) Proof. See Anufriev et a. [26]. Proposition 2.1 describes the equiibrium price dynamics in the sense that at each period t Warasian equiibrium is achieved on our two-asset market. It is straightforward to see that the equiibrium return must satisfy: i r t = w t 1,i (x t,i x t 1,i + x t,i x t 1,i e t ) i w, (3) t 1,i (x t 1,i x t,i x t 1,i ) 4 Documents de Travai du Centre d'economie de a Sorbonne

7 if the rescaed return is defined by (1). Note that equation (3) expicity specifies the return r t conditionay to the information set at period t 1, if and ony if we impose additiona assumptions: both the demand x t,i and the dividend yied e t must be independent of the current price eve p t. The simpest assumption about dividends one can suggest to make the mode in (2) tractabe, is that the dividend yied is an iid non-negative stochastic process. Foowing Chiarea and He [21] and Anufriev et a. [26], we stick to this assumption, though we are aware of the constraints it imposes. Dividends in our economy are deprived of their own dynamics, but foow the risky asset price. Roughy speaking, the amount of dividends avaiabe is supposed to automaticay adapt to the fuctuations of the price eve, so that the mean dividend yied remains unchanged. In rea ife dividends are paid by stock issuers and so depend on companies profits and decided payout ratios. If the suppy of the risky asset is fixed, one can hardy expect a perfecty inear dependence between average dividends and prices, though a positive reationship between them does exist. However, for the purposes of our paper, the iid assumption for the dividend yied is sufficient. So far, nothing has been said about the way agents determine the desired proportions of investment in the risky asset. The MIS hypothesis, studied in this paper, impies that some investors do not trade at a time periods and remain passive. During the period, when some investor is out of the market, his share of investment in the risky asset is no onger a resut of his decisions but a consequence of price and weath movements, independent of his wi. The foowing proposition specifies the way investment shares evove. Proposition 2.2. Let x k t,i be the share of investment in the risky asset of investor i, who actuay participated in the trade k periods ago, k = 1,...,h with h being his investment horizon. The investment share verifies the foowing recurrent reationship: Proof. See Appendix. x k t,i = x k+1 t 1,i (1 + r t) 1 + x k+1 t 1,i (r t + e t ) (4) At the period when investor i readjusts his portfoio, his demand for the risky asset x t,i is determined according to some investment function. In this paper, we suppose that investment functions are given as the dependence of the share of weath, invested in the risky asset, on the beiefs about future gains. We aso suppose that investment functions are deterministic and do not change over time for the same investor 1. The beiefs are based on the past observations of prices and dividends, without any private information that coud be used to forecast future returns. Moreover, each investment function is supposed to be independent of the current weath, which is a natura assumption in the CRRA framework. So investor i s function reads: x t,i = f i (r t 1,...,r t Li,e t 1,...,e t Li ) (5) where L i is the maximum ag for historica observations used by the agent i, which can be finite or infinite. 1 Note that this does not excude functions, corresponding to investment strategies that evove according to predefined rues. 5 Documents de Travai du Centre d'economie de a Sorbonne

8 In this paper we wi in particuar focus on the case of preferences that corresponds to the maximization of the mean-variance CRRA expected utiity of weath 2. Let us suppose that investors, possiby operating over different time scaes, maximize a mean-variance expected utiity: max x t,i { E t 1,i (W t+h,i ) γ i 2W t,i Var t 1,i (W t+h ) with operators E t 1,i ( ) and Var t 1,i ( ) standing for the beiefs of agent i about the mean and variance given the information at time t 1. The information set of period t 1 incudes the prices of the risky asset and the dividends at time t 1 and earier. The coefficient γ i is a positive constant that measures the risk aversion of investor i. The time horizon of decision taking, denoted h, corresponds to the period of time when investor i does not readjust his portfoio. The number of units of risky asset in investor s possession remains constant over [t;t+h], whie the share of investment in the risky asset may evove. We assume that dividends, paid by the risky asset during this period, are accumuated on the bank account, yieding the risk-free rate. Proposition 2.3. The soution x t,i of the maximization probem (6) is approximatey given by: Proof. See Appendix. x t } (6) [ h ] E t 1,i (e t+k + r t+k ) [ h ] (7) γ i Var t 1,i (e t+k + r t+k ) Chiarea and He [21] show that the expression (7) with h = 1 aso emerges as an approximative soution in the maximization probem with the power utiity function. This approximation, however, consists in a discretization of a continuous-time process with Gaussian increments and thus it can be far from the rea soution for non-infinitesima time units. So we prefer to work with mean-variance maximization directy. Aternativey, an investment function of the form (7) coud be set on an a priori basis since it describes the behavior of a mean-variance investor with constant reative aversion to risk. Notice that if the return process is iid, E t 1 [y t,t+h ] = h E t 1 [r t+1 + e t+1 ] and Var t 1 [y t,t+h ] = h Var t 1 [r t+1 + e t+1 ]. This ensures that if, in addition, risk aversion is homogeneous for investors at a scaes (γ i = γ), the demand for the risky asset does not depend on the investment horizon. We maintain the assumption of homogeneous risk aversion throughout this paper. In equation (7) the share of weath to be invested in the risky asset depends excusivey on the beiefs of agents about future yieds. In the heterogeneous agents iterature these beiefs are based on historica prices of the risky asset up to a certain ag. Foowing Chiarea and He [21], we do not incude p t in the information set for beiefs formations in order to avoid unnecessary compexity. Nevertheess, in the MIS case the aggregate demand on the risky asset naturay depends on the current price eve. Indeed, suppose that the previous date, when 2 Most of our resuts are vaid aso in the case of the genera investment function (5), not necessariy representing the beiefs about mean and variance. This wi be speciay indicated further in the paper. 6 Documents de Travai du Centre d'economie de a Sorbonne

9 investor i participated in the trade, was t k and that at this date the share of weath x t k,i he invested in the risky asset was determined according to (5). Then it foows from (4) that his current investment share x k t,i depends on the historica returns and dividend yieds up to the ag L i + k 1, but aso on the current return and the dividend yied, which are unknown before the trade at date t. So equation (3) does not expicity specify the dynamics of the risky return. In the foowing section we study the dynamics of the price and weath in the mode with one scae, which is a particuar case of the mode, introduced in the previous section. We further refer to it as the benchmark mode. We extend the anaysis of Anufriev et a. [26] in severa aspects, aso important in the MIS case, studied ater. 3 Equiibria in the One-Scae Mode with Bounded Rationaity As we have mentioned before, in the one-scae case, equation (3) competey and expicity describes the dynamics of the return on the risky asset under the market cearing condition. By specifying the demand function, one can determine the equiibrium price and return. This equiibrium dynamics was earier studied in Anufriev et a. [26], who repace the actua dividend yied by its mean and work with the so-caed determenistic skeeton of the system. In the deterministic case the (rescaed) return is constant: r t = r. The authors prove that two types of equiibria are possibe: either a singe agent survives 3, or many agents survive, but in both cases the equiibrium share of investment in the risky asset and the steady growth rate of its price are determined in a simiar way. They must satisfy the reationship, which is easiy obtained from (2) for a singe-agent case, when we set x t = x t 1 for a t. This reationship is caed the Equiibrium Market Line (EML) and reads: x = r r + ē where ē is the mean dividend yied. The demand functions of investors depend on a singe variabe and are of the form: x = f(r t 1,...,r t L ) = f(r,...,r) = f(r) (9) The equiibrium points are determined as the intersections of the demand curve f(r) and the EML. It is shown that, if mutipe agents survive, their demand functions must a intersect the EML at the same equiibrium point. Stabiity conditions, depending on the properties of derivatives of f i ( ) with respect to returns at different ags, are estabished. We refer the reader to the origina paper of Anufriev et a. [26] for further detais. In our approach, the main difference is that we are interested in the stochastic properties of the return process. In particuar, we estabish anayticay, under what conditions the dynamics of returns is simpe (iid) and when it dispays interesting dynamic patterns (conditiona heteroscedasticity and/or 3 i.e. his share in the tota weath does not decrease to zero in infinite time (8) 7 Documents de Travai du Centre d'economie de a Sorbonne

10 seria correations). In our view, this type of approach is appropriate for the study of boundedy rationa behavior of agents, whose investment functions are based on beiefs about mean and variance of the return process. This point is expained further. For the case of mutipe agents with heterogeneous investment functions, Anufriev et a. [26] determine which form of the demand function dominates the others. For exampe, if a trend-foower (investor who strongy extrapoates past returns) meets a fundamentaist (investor, whose demand function is independent of the price history), we can predict which of them survives, depending on the respective form of their investment functions. Under some conditions, the trend-foower outperforms the fundamentaist and survives. A striking feature of the mode is that equiibria are possibe for amost any, and even competey senseess, demand functions and can even be stabe. The probem here is with bounded rationaity. More precisey, it is important to what extent the rationaity is bounded. In Anufriev et a. [26] and Chiarea and He [21], investment functions are given a priori, and though they formay depend on the beiefs of agents about the mean and variance of future returns, there are no constraints on how these beiefs shoud be reated to the true quantities. Bounded rationaity means that agents may not know the true mode. But in equiibrium, when the return on the risky asset is supposed to be constant, it is hard to admit that the beiefs have nothing to do with reaity. Besides, the stabiity of such equiibria hardy makes sense from the economic point of view, since agents woud have incentives to change their strategies, if they were aowed to. In Brock and Hommes [1998] agents are aowed to switch between strategies, according to the profits they yied in the past. The agents can thus be caimed to be proceduray rationa, because they try to rationay choose strategies according to some criteria. In our case, a more exact definition of procedura rationaity can be hepfu to study the mode anayticay. We restrain the cass of admissibe investment functions, consideraby reducing the possibiities for non-rationaity of economic agents, without necessariy imposing rationa expectations. Investment functions basicay define how agents beiefs about future returns are formed, i.e. they are concise descriptions of the outputs of the beiefs-making procedure. The rationaity of such procedure can be tested in some simpe reference case, where the outcome of the procedure is expected to correspond to the rationa behavior. In our case of mean-variance investor, we require that for the iid returns beiefs about mean and variance of the process shoud be unbiased. This is formaized in the foowing definition. Definition 3.1. An investment function of the form: x t,i = f i (E t 1,i [y t,t+h ],Var t 1,i [y t,t+h ]) (1) is caed proceduray rationa if the beiefs E t 1,i (y t 1,t+h ) and Var t 1,i (y t,t+h ) about the mean and variance of the future tota returns are unbiased estimates of these quantities with finite error, if the true process y t,t+1 is iid. This definition is an adaptation of Simon s procedura rationaity 4 to our 4 Behavior is proceduray rationa when it is the outcome of appropriate deiberations. Its procedura rationaity depends on the process that generated it [Simon, 1976, p.131]. 8 Documents de Travai du Centre d'economie de a Sorbonne

11 context. It basicay states that, if previous observations of returns dispay no non-trivia dynamic patterns, the beiefs about mean and variance of investors shoud have no systematic error. Note that in no way we state that returns shoud actuay foow an iid process. We ony describe the behavior of the investment function in this hypothetica case, in order to impose some constraints on the reasonabiity of the decision taking procedure used by investors. Note that our definition does not contradict to the concept of bounded rationaity, but it requires some moderate degree of consistency in investors beiefs. Proceduray rationa investors can actuay be trend-foowers or contrarians. Consider, for exampe, the foowing specifications for the beiefs about the mean of future returns: E t,i (y t+1 ) = c i + d i E t,i (y t+1 ) = 1 d i L L y t k y t k + d i (A) y t k (B) (11) The function of the type, anaogous to (11A), is used in Chiarea and He [21] to represent the behavior of heterogeneous investors. Here c i is some constant that represents the risk premium, required by the investor, and d i is a behaviorist parameter, which specifies, how investor i extrapoates the performance of the risky asset over recent periods. If d i =, the investor is fundamentaist, if d i > he is a trend-foower, otherwise contrarian (chartist). It is easy to see that this specification does not correspond to our definition of the procedura rationaity, uness simutaneousy c i = and d i = 1. The function (11B) aso aows for the extrapoation of the recent returns via the parameter d. If < L, positive d i corresponds to the trend-foowing. But this function verifies our condition for procedura rationaity: in the iid case the expectation of the difference in the short-term and the ong-term mean is nu. Suppose that investors preferences are described by the mean-variance function of the form (7), satisfying definition 3.1. Having restrained the set of admissibe investment functions, we turn to the study of the price dynamics in the benchmark mode. In the foowing theorem we estabish the conditions that must be verified by the investment function to ensure simpe dynamics of the returns, which can be associated with some steady growth trajectory. It states that the assumption of investors rationa expectations is equivaent to the iid dynamics of returns. Theorem 3.2. In the benchmark mode with homogeneous proceduray rationa agents the return process can be iid with finite mean and variance if and ony if investors have rationa expectations. In this case the mean and variance of the return process are uniquey defined by the mean dividend yied and investors risk aversion. Proof. The homogeneity of agents means that they a have the same investment functions and, in particuar, the same risk aversion γ i = γ. In the benchmark mode they aso use the same information, so x t,i = x t,j, t,i,j and we can drop the second subscript. Thus this case is anaogous to a singe-agent mode with a representative agent. Simpifying (2), it is straightforward to see that the 9 Documents de Travai du Centre d'economie de a Sorbonne

12 returns do not directy depend on the weath dynamics, since we have: r t = x t x t 1 + e t x t x t 1 (1 x t )x t 1 (12) If r t is an iid process, then r t is independent of the returns history r t 1,r t 2,..., but it is aso independent of x t,x t 1,... since the atter depend ony on past returns. Consider the stochastic process r t t 1 of returns, conditiona to the information at period t 1, which is defined as the set I t = {r t 1,r t 2,...;x t,x t 1,...}. It foows from the above that this process is aso iid. The quantities x t t 1 and x t 1 t 1 are both deterministic since the investment function at time t depends ony on returns at time t 1 and earier. So the conditiona mean and variance of returns are: E t 1 (r t ) = x t x t 1 + ēx t x t 1 (1 x t )x t 1 (13) x 2 t Var t 1 (r t ) = σe 2 (1 x t ) 2 (14) with ē and σ 2 e the mean and variance of the dividend yied process respectivey (both are supposed to be constant). Note that here the operators E t ( ) and Var t ( ) no onger refer to the agent s beiefs, but to the mathematica expectation and variance of random variabe. We have shown that the process r t t 1 is iid. Then it foows from (13) that x t = x t 1 = x and equation (12) simpifies to: x r t = e t 1 x (15) The investment function f(r t 1,...,r t L ) takes the vaue x with probabiity 1 for a vaues r t 1,...,r t L drawn from an iid process if and ony if it is a constant function in any domain where the vector r t 1,...,r t L takes vaues with non-zero probabiity. Since the return dynamics, given by (15), is iid, procedura rationaity impies that the beiefs of investors are unbiased: x E t 1 [r t+1 + e t+1 ] =ē 1 x + ē = ē 1 x [ ] et+1 σe 2 Var t 1 [r t+1 + e t+1 ] =Var t 1 1 x = (1 x ) 2 (16) Then, according to (7) with h = 1, the investment share satisfies: ē 1 x x = (17) σe γ 2 (1 x ) 2 From (17) we obtain a unique soution for x 5 : x = ē γ σ 2 e + ē (18) 5 Anaoguous computation in terms of not-rescaed variabes gives x = E t 1 [ε t+1 ](γ Var t 1 [ε t+1 ] + E t 1 [ε t+1 ]) 1, which is sighty different from (18) because of the first order approximation. This difference is of no incidence in our context. 1 Documents de Travai du Centre d'economie de a Sorbonne

13 This proves that if returns are iid, then the investment share x, computed from (18), uniquey determines the mean and variance of the process r t t 1 (or, in other words, the necessary conditions for the iid return dynamics). It is easy to see that the soution we derived corresponds to the case where investors have rationa expectations. It can be shown straightforwardy, that these conditions are aso sufficient. It suffices to pug the constant x in the equation (12) for returns and then verify that the expectation and variance of returns are constant and given by (13) and (14) respectivey. An important consequence of theorem 3.2 is that, in the benchmark mode with homogeneous proceduray rationa investors, returns on the risky asset never have simpe iid dynamics, uness the investors have rationa expectations. Note that equation (14) describes conditiona voatiity dynamics in the mode. It foows from (14) that for < x t < 1, conditiona variance aways increases with x t. If the investment function depends positivey on the historica mean of returns and negativey on their historica variance (which is an appropriate assumption in a proceduray rationa context), then the conditiona variance is a decreasing function of historica variance and increasing function of historica returns. At the same time, voatiity has the same memory as the squared share of investment in the risky asset, which is determined by investors beiefs. If the atter are adjusted sowy, then voatiity aso adjusts sowy. Now consider the return dynamics in a more genera case, when homogeneous agents having arbitrary (not necessariy boundedy rationa) investment functions of the form x t = f(r t 1,...,r t L,e t 1,...,e t L ). The stochastic process (12) for the return dynamics is non-inear. We study the properties of its first-order Tayor inearization in the neighborhood of the expected return, r t k = r for a k, with arbitrary vaues of e t k. We denote ẽ t = e t ē and r t = r t r the deviations of dividend yied and return from their average vaues. We aso denote f k the first derivative of f( ) with respect to r t k for k = 1,...,L. The form of the return process is given by the foowing theorem. Theorem 3.3. In the benchmark mode with homogeneous agents, if the return process is covariance stationary, it satisfies: r t = x 1 x ē + r t L+1 r t = a k r t k + v t ẽ t v t = x L 1 x + b k r t k (19) 11 Documents de Travai du Centre d'economie de a Sorbonne

14 with: a 1 = f 1[1 x(1 ē)] x(1 x) 2 a k = f k [1 x(1 ē)] + f k 1 ( x 1) x (1 x ) 2, k {2,...,L} f L a L+1 = x(1 x) Proof. See Appendix. f k b k = (1 x) 2, k {1,...,L} x = (2) Equations (19) can be written in the equivaent form: L+1 r t = u 2 t = a k r t k + σ e ( u t + L b 2 k r t k i,j {1,...,L} i j x ) ε t 1 x b i b j r t i r t j (21) with ε t a standardized independent white noise. This stresses the ARCH-ike nature of the returns process. Note that its mean is described by an expression, equivaent to the definition of equiibrium on the EML in [Anufriev et a., 26]. Now we can turn to the case with heterogeneous agents, i.e. the case when x t,i are determined in a different way by each investor. Again we restrict the investment functions to mean-variance and proceduray rationa. Theorem 3.4 shows that the simpe iid dynamics does not appear genericay if investors are heterogeneous. Theorem 3.4. In the benchmark mode with heterogeneous proceduray rationa agents the return process can be iid with finite mean and variance ony if the aggregate share of weath invested in the risky asset is constant. In this case the mean and variance of the return process are proportiona to the mean and variance of the dividend yied. Proof. See Appendix. Basicay this theorem says that if the aggregate share of investment in the risky asset is subject to stochastic shocks or fuctuations, the return dynamics is amost surey not trivia and dispays dynamic patterns. The situation, when the aggregate investment function is constant and returns are iid, can arise ony when the dependence of the individuas proceduray rationa investment functions on the past returns is not characterized by prevaiing patterns. More precisey, individua deviations ν t,i = x t,i x from some constant investment share x, weighted by the weath portions of agents ξ t,i, are eiminated by aggregation with probabiity one: N P( ξ t,i ν t,i = ) = 1 i=1 12 Documents de Travai du Centre d'economie de a Sorbonne

15 for a t. For this condition to be true, some form of the aw of arge numbers must be satisfied and, moreover, the expectation of ν t,i, conditiona on past returns, must be constant. This is improbabe in the situation, when a investors base their expectations on the same vector of reaized past returns and this vector is not constant. 4 Equiibria with Mutipe Investment Scaes In the previous section we considered the case when investors have the same investment horizons, but possiby different investment functions. Now we come back to th MIS hypothesis and study another source of heterogeneity, reated to investment horizons. Now assume that there exist H investment scaes with portfoio readjustment periods h = 1,...,H time units, so that each agent has a characteristic investment scae that does not change. Suppose that within each investment scae investors are homogeneous, i.e. have the same specifications of demand function. Finay, suppose that at each date the weath of investors, having the same investment scae, is distributed so that a constant part of this weath, equa to 1/h beongs to the investors, rebaancing their portfoios at the current date. The atter assumption does not necessariy impy that the weath can be redistributed between different groups of investors in a given period. Rather, it means that there is a arge number of investors, going in and going out, and they have random dates of intervention on the market but fixed frequencies of trades. So the composition of each cohort of investors may change, but its average share of weath remains constant. Under these simpifying assumptions, we can aggregate a investors, acting at the same scae h, and repace them by a representative agent, whose share of weath, invested in the risky asset, satisfies: x t,h = 1 h h 1 x k t (22) Equations (2), describing the dynamics of the system, are sti true, but now the subscript i corresponds to the investment scae and the weath w t,i is the aggregate weath of a cass of investors, having the same investment horizon. In section 2 we derived equation (4) that describes the evoution of the share of investor s weath, invested in the risky asset, when he does not trade. Then the compete system of equations, describing the dynamics of risky return, reads: k= H r t = w H t 1,h (x t,h x t 1,h ) + e t x t,hx t 1,h w t 1,h H w t 1,hx t 1,h (1 x t,h ) x t,h = 1 h x k t,h = h 1 k= x k t,h x k+1 t 1,h (1 + r t) 1 + x k+1 t 1,h (r t + e t ) (23) As noted above, an important feature of (23) is that it describes the return dynamics ony impicity, because the investment share for a but the shortest 13 Documents de Travai du Centre d'economie de a Sorbonne

16 scaes inevitaby depends on current return. The reation between the price and the dividend process becomes non-inear and compicated, because it incudes previous dividends. For the genera equation of price dynamics, we can prove that: Theorem 4.1. Whatever the number of scaes H, there aways exists at east one positive market cearing price for which the return r t satisfies (23). Proof. See Appendix. It is important to specify conditions, under which the muti-scae dynamics does not degenerate, that is the portions of weath, hed by the agents, investing at each scae, do not tend to zero as time tends to infinity. More precisey, denote ξ t,h the portion of weath that beongs to investors of type h. Definition 4.2. The MIS dynamics, described by equation (23), is caed nondegenerating, if for any investment scae h such as ξ,h > we have : when t approaches infinity. P(ξ t,h = ) =, In the foowing theorem we estabish the necessary and sufficient conditions that provide for the non-degenerating dynamics in the MIS system. Denote g t,h the growth rate of weath of investors of type h at time t: g t,h = w t,h w t 1,h = 1 + x t 1,h (r t + e t ). We suppose that the stochastic process n(g t,h ) is covariance-stationary. Furthermore, we suppose that it verifies the foowing conditions on its memory: N 1 {Cov (n(g t+i,h ),n(g t+j,h ))} {i=1,...,n,j=1,...,n} 2 C (24) for a positive N and some finite C. This technica condition, impying that n(g t,h ) is a stochastic process with bounded spectra density, ensures that the average growth rates of weath converge amost surey to their expectation as time tends to infinity. This resut is proved in Ninness [2]. Theorem 4.3. The mutipe investment scaes dynamics, described by equation (23), is non-degenerating if and ony if for any h: Proof. See Appendix. E[n(g t,i )] = E[n(g t,j )], i,j {1,...,H} To interpret the theorem, notice that the og growth rate of the weath is approximatey equa to the product of the tota return on the risky asset and the share of weath, invested in the risky asset at the previous period. Thus, for the mode to be non-degenerating, investors shoud either have the same average share of investment in the risky asset, or ower investment shares shoud be compensated by positive correation of the investment share with future return. 14 Documents de Travai du Centre d'economie de a Sorbonne

17 A particuar case of the non-degenerating system is non-predictive equa-in-aw investment shares: x t,i L =xt,j i,j {1,...,H}, t Cov (x t i,h,r t ) =, h {1,...,H}, i,t. (25) Note that in MIS system the existence of autocorreations in returns impies correation of the investment shares with the future returns. Moreover, the atter is higher for investors at onger scaes, because at each time period there are more passive investors, whose investment shares depend on past returns, even if eementary investment functions are constant. Thus condition (25) is reated to the absence of seria correations in returns. By anaogy with the one-scae case, we anayze the equiibrium dynamics of the system (23). First et us study the mean dynamics, supposing e t = ē. The foowing theorem shows that there exists an equiibrium path r t = r that soves the deterministic anaog of (23). Theorem 4.4. The dynamic system (23) with e t = ē has a unique equiibrium soution with constant return: r = x 1 x ē, f h ( r,..., r ) = x (26) Proof. Suppose that the system has some equiibrium soution r = r. This impies: H w t 1,h [ x t,h x t 1,h + ē x t,h x t 1,h ] H w = r < (27) t 1,h x t 1,h (1 x t,h ) Besides, it is easy to notice that, for any h, investment in the risky asset is constant, because the investment functions depend ony on the past reaizations of returns and dividend yieds, equa to r and ē respectivey: x t,h = f h ( r,..., r,ē,...,ē) = x h. At the same time conditions (25) impies that average investment shares are equa for a types of investors. Thus the trajectories of weath satisfy: Thus equation (27) simpifies to: which is equivaent to: w t 1,h = w,h [1 + x( r + ē)] t 1. ē x 1 x = r, x = r r + ē. (28) We need to verify that (28) is compatibe with the muti-horizon dynamics of the investment shares, characterized by passiveness of a part of agents at some time periods. Reca that the investment in risky asset of each type of agents h is the mean of investments of agents that readjusted their portfoios with,...,h 1 periods ago. But whenever the readjustment takes pace, 15 Documents de Travai du Centre d'economie de a Sorbonne

18 the investment share, depending on agged returns and dividend yieds, aways takes the same vaue x h. At the next period after portfoio readjustment the investment share of the passive investor becomes: x h (1 + r) 1 + x h ( r + ē) We define the function g = R R as: g(x) = x(1 + r) 1 + x( r + ē) (29) and g k (x) as a k-times composition of function g( ), that is g g... g(x), with g (x) defined as g(x) = x. Then for any h we have: Now notice that r x h = 1 h h g k ( x h) (3) k= r g( r + ē ) = r r + ē, which impies that x h = r+ē satisfies equation (3). This proves that if the equiibrium return exists, it satisfies: and thus is uniquey defined. r = ē x 1 x, We can now study the properties of the stochastic process for the risky returns and compare the resuts with those, obtained for the one-scae case. As before, we wi proceed by the inearization of the dynamic system. Define the foowing function F : R t 1 R t 1 R R R: F(r 1,...,r t 1,e 1,...,e t 1,r t,e t ) = H w t 1,h [x t,h x t 1,h + e t x t,h x t 1,h ] (31) H w r t t 1,hx t 1,h (1 x t,h ) with x t,h defined as in (23). The foowing theorem describes the equiibrium dynamics in the neighborhood of the soution of the deterministic anaog of the system. Theorem 4.5. In the mode with homogeneous rationa agents and mutipe investment scaes, the return process is approximatey described by: r t = r + ˆr t + V ẽ t, ˆr t = V t = H 1 H 1 A k ˆr t k + V t ẽ t, B kˆr t k (32) 16 Documents de Travai du Centre d'economie de a Sorbonne

19 where: A k = a k b k (1 c)(1 + r ) k, B k = x(1 2 x )(cb k a k ) (1 x)(1 + r ) k+1 (1 c), V = x 1 x, H h k 1 a k = ξ,h, h b k = c = h=k+2 H h=k+1 H h k h ξ,h, h 1 h ξ,h. Proof. See Appendix. The resut of theorem 4.5 shows that the return dynamics in the mutiscae mode with rationa investors is very cose to the one-scae dynamics in the rationa expectations case, the ony difference being the term ˆr t. It represents the deviation from the hypothetica trajectory of returns, that woud be reaized in a one-scae market, and can be interpreted as error correction term. Note that there is no constant in the voatiity of the disturbance term, which means that the correction term either vanishes or expodes, depending on the vaues of coefficients A k, B k and the variance of ẽ t. We wi study its behavior for pausibe vaues of parameters in the foowing section. Note that the dynamics in the mutiscae case is consideraby different from the genera one-scae case, described by theorem 3.3. There the terms, containing seria correations and heteroscedasticity, are not vanishing, whie in the mutiscae rationa case their presence is tempora after a shock, in the absence of which they competey disappear from the return dynamics. Theorem 4.5 refers to the case, when investors demand functions at the times of portfoio readjustment are trivia: investment shares are constant at the eve, corresponding to the rationa equiibrium, which coincides with the one-scae equiibrium. In practice, investment decisions may depend on the historica returns, so the framework of procedura rationaity woud be more adequate for modeing. One can estabish a genera anaytica representation of the return dynamics in this case. Indeed, equations (A-9), (A-11) and (A-12) in the proof of theorem 4.5 (see Appendix) sti hod, but instead of (A-1) we need to adopt a genera form for the investment functions, as in theorem 3.3. However, in our view, such genera representation woud be of itte practica vaue. Instead, using simuation, we expore the return dynamics, corresponding to concrete styized exampes of investment functions. This issue is addressed in the foowing section. 17 Documents de Travai du Centre d'economie de a Sorbonne

20 5 Simuation Study We determined the equations of the risky return dynamics in the case of the market, popuated with rationa participants, acting at one and severa investment horizons. We aso estabished the framework for the study of the proceduray rationa investment, that can incorporate behaviora patterns, such as trend extrapoation and contrarian strategies. Our goa in this section is to expore the empirica properties of the return series, generated by different versions of our mode, and to associate the properties of the mode with the styized patterns, observed on rea market data: contrarian returns, trend formation and conditiona heteroscedasticity. From observation of (32) it is cear that introducing mutipe scaes changes the way, in which the dynamic system for the risky return reacts to shocks. These shocks coud be of competey exogenous or of behaviora nature. We wi first study the case when, aong with the norma disturbance term, interpreted as dividend yied, the mode is occasionay perturbated by exogenous shock on returns, unreated to the investment functions. Such abnorma returns can refect deviations from market cearing equiibrium at some time periods. Returns trajectories are simuated for a market with five horizons, where abnorma returns occur at random periods, on average once per 5 trades. We are interested in the vaues of coefficients A k and B k, that determine the way the shock at period t is reverberated at future dates. Note that in a one-scae mode such shocks have absoutey no incidence on future returns. The abovementioned coefficient depend on a k,b k and c, that characterize how initia weath is distributed among investors. 18 Documents de Travai du Centre d'economie de a Sorbonne

21 Figure 1: Shocks to returns, -shape weath distribution (a) Mode Parameters ξ (k) 2 4 k a(k) b(k) c(k) c k A(k) B(k) k (b) Reaction to Exogenous Shocks.1 r t.6.4 ρ( r t, r t ).6.4 ρ( r t, r t ) t 1 x 1 3 V t ρ(v t ẽ t, V t ẽ t ) ρ( V t ẽ t, V t ẽ t ) t Mode with H = 5, where initia weath is distributed according to discretized β(2, 2) distribution, e N(.3,.2 2 x ), x =.75. Gaussian shocks with variance 2 (1 x ) 2 σǫ 2 are appied to 1 the r t series at random dates with frequency, i.e. on average every 5 points. Autocorreations are estimated on a 1 - periods simuation 1 H path. 19 Documents de Travai du Centre d'economie de a Sorbonne

22 Figure 2: Shocks to returns, -shape weath distribution (a) Mode Parameters ξ (k) 2 4 k a(k) b(k) c(k) c k A(k) B(k) k (b) Reaction to Exogenous Shocks 5 x 1 3 r t.4.2 ρ( r t, r t ).4.2 ρ( r t, r t ) t 6 x 1 3 V t ρ(v t ẽ t, V t ẽ t ) ρ( V t ẽ t, V t ẽ t ) t Mode with H = 5, where initia weath is distributed according to discretized β(2, 2) distribution, e N(.3,.2 2 x ), x =.75. Gaussian shocks with variance 2 (1 x ) 2 σǫ 2 are appied to 1 the r t series at random dates with frequency, i.e. on average every 5 points. Autocorreations are estimated on a 1 - periods simuation 1 H path. 2 Documents de Travai du Centre d'economie de a Sorbonne

23 Figure 3: Shocks to returns, weath shares increase with investment horizon (a) Mode Parameters ξ (k) 2 4 k.2 a(k) b(k) c(k) c k A(k) B(k) k (b) Reaction to Exogenous Shocks r t ρ( r t, r t ) ρ( r t, r t ) t 1 x 1 3 V t ρ(v t ẽ t, V t ẽ t ) ρ( V t ẽ t, V t ẽ t ) t Mode with H = 5, where initia weath is distributed according to discretized β(2, 2) distribution, e N(.3,.2 2 x ), x =.75. Gaussian shocks with variance 2 (1 x ) 2 σǫ 2 are appied to 1 the r t series at random dates with frequency, i.e. on average every 5 points. Autocorreations are estimated on a 1 - periods simuation 1 H path. 21 Documents de Travai du Centre d'economie de a Sorbonne