Economics Working Paper 55 Heterogeneous Beliefs, Wealth Accumulation, and Asset Price Dynamics* Antonio Cabrales Universitat Pompeu F and Takeo Hoshi U niversity oí California, Keywords: Adaptive Dynamics, Wealth Dynamics, Portfolio Choice, Arch, Star. Journal 01 Economic Literature classification: GIO, G12. • The authors would like to thank Valentina Corradi, Dean Foster, Daniel Friedman, Alex Kane, Makoto Saito, Joel Sobel and a seminar audience at UCSD for their helpful comments. Abstract A model of asset markets with two types of investors is developed and its dynamic properties are analyzed. "Optimists" expect on average higher returns on the risky assets than "pesaimists" do. The stochastic procesa íor equilibrium asset return changes over time as the distribution oí wealth between the two types oí investors changes. In the long run, the share oí wealth held by one type oí investor may become negligible, but it is a180 poesible íor both types to co-exist, depending on the parameter values of the model. Relations between this model and sorne econometric models with time varying parameters, such as the ARCH (Autoregresaive Conditional Heteroskedasticity) model and the STAR (Smooth Transition Autoregresaive) model, are examined. The dynamic properties oí another model, regarding investors who use strategies that are a bit more complex, are al80 analyzed. "Fundamentalists" believe that the asset returns íollow a procesa that is solely determined by íundamentals and "contrarians" assume the market is wrong and choose a portfolio that is exactly opposite of the market portíolio. Again, depending on parameters, both types can co-exist even in the long runo 2 1. INTRODUCTION Recent studies in fmance, which inelude Shiller (1984), Black (1986), Frankel and Froot (1988, 199Oa, 199Ob), and De Long, Shleifer, Summers, and Waldmann (DSSW hencefonh) (1990) among others, stressed the importance of heterogeneous beliefs in asset markets. These studies show that models with heterogeneous investors are useful in explaining sorne empírical puzzles that cannot be explained by a model witb a representative investor. This paper studies a similar model witb hetcrogeneous beliefs to analyze the asset price dynamics. We consider an asset market which includes two types of investors, who follow two different rule of tbumb strategies. Depending on the relative success of each strategy, the proportion of total wealth held by each type of investor fluctuates. If strategy A is more successful tban the other strategy B, for example, tbe investors who use strategy A increase tbeir wealth more than tbe other investors, invcst more in the asset, and inCl'ease their influence on asset pricing. Thus the changes in the distribution of wealtb between tbe two types of invcstors affect tbe uset prices, which in turn influence tbe relative suceess of each StrBtegy and again change wealtb distribution. Profits, which are random, are use<! for buying back tbe shares of the risky uset to create capital gains for tbe ownen, or for paying out dividends to create income gains. Thus random profit shoca influence tbe dynamics of tbe asset price and wealtb distribution. The uset price and wealth distribution themselves also bccome stochastic processes. This paper studies me limiting behavior of these stochastic processcs. Under some paramctcr values, we show that the limiting behavior of those dynamics does not ciepend 00 the initial condition. The most important empírical implication of our model is smootb shifts of the 3 price dynamics. Both the conditional mean and the conditional variance of the asset price change over time reflecting changes in the wealth distribution between two types of investors. For sorne periods, strategy A may do bettcr than strategy B and enhance its influence. Following sorne random shocks to profits, strategy B may become more profitable, and the proportion of wealth held by investors who use stratcgy B will inercase. Following sorne other shocks, stratcgy A may again become more profitable and regain its influence. In this way, wealth distribution fluctuates over time and asset price dynamics show smootb shifts betwccn two extreme regimcs: one where all tbe wealth is owned by investors who use strategy A and the otber where all tbe wealth is owned by investors who use strategy B. Altbough these extremes never happen if tbe limiting distribution of tbe proportion of wealth is ergodic, tbe price dynamics still can show substantial variation over time without reaching an extreme. Thos our model is useful in explaining an apparent empirical regularity in financial markets: namely time varying conditional variance of asset retums. Tune varying conditional variance bu becn dcrected by many researchers who estimaced ARCH (Autoregressive Conditional Heteroskedasticity) models to financial data. Bollerslev, Chou, Jayaraman, and Kroner (1990) surveyed numcrous successful applications of ARCH models in Ílnance. The rOl' modcl witb the representarlve agent docs not usually have strong implications the dynamics of conditional variance of asset retums. Thus, in ordcr to intcrpret empirical success of ARCH models in finance, the standard model has to assume ARCH at the fundamental level, fOl' example in tbe process of cash flow. The modcl in this papee can explain time varying conditional variance without assuming ARCH at the fundamental level Tbe model also implies that the mean growth rate of uset price changes over time, ando 4 changes in the mean are related to changes in the conditional variance. Thus a type of ARCH models called the ARCH-M model, which is used to capture shifts in the conditional mean associated with changes in the conditional variance, becomes especially relevant for our model. Our model is also useful in motivating another statistical model with time varying parameters called STAR (Smooth Transition Autoregressive) model. The STAR model was applied 10 some aggregate variables like GNP and industrial production by Anderson and TeIisvirta (1991). A STAR model assumes that a process is a weighted average of the two distinct AR processes and the weight changes over time. Our model also implies thal the usel price process shifts between two extremes, although the variable that determines the weight in our model (proportion of wealth) enters the price process in a complex way. There are several studies of the dynamics of assel prices in the uset market with heterogeneous agents. DSSW (1991) considers the wealth accumulation by two typeS of investors. noise traders and sophisticated investors. The noise traders have random and biased forecuts of the rate of retum and its v~ance, whereu the sophisticated investors have the correet forecasts. They show, under some parameter values. that the noise traders' wealth may grow faster than that of the sophisticated investors and eventually the noise traders may domínate the market One problem of DSSW (1991) is that they ignCR the noise traders' influence 00 the uset price. which is the most important point made by models with noise traders. such as DSSW (1990). Our model explicidy studies the way the heterogeneous beliefs intluence asset prices and examines the price dynamics. Another important difference between our paper and DSSW's research is that we do not assume the presence of sophisticatec1 investors who have rational expectations and maximize their 5 expected utility. Although the agents in our rnodel use ponfolio strategies that rnaximize an expected utility function under sorne assurnptions about beliefs, they do not have rational expectations. Thus the agents in this paper deviate from the standard rational consumen in economics. In this sen se, our approach is similar to that followed by Blume and Easley (1992). They consider the dynamic process of an asset maricet with heterogencous investors, which is similar 10 ours. Each type of investor uses different portfolio rules, and the market eventually selects the most "6t" rules, in the sense that wealth held by such investors grows faster than that of the other investors. One difference between our model and Blome and Easley (1992) is the formulation of asset retum. Blume and Easley (1992) assume an exogenous probability distribution over the possible pay-out oC the asset. Thus the current price of the asset does not influence the rate of retum on an asset that was bought last period. In other words, the assets in their model can have only incomc gains and not capital gains. Our model includcs both incomc gains and capital gains of asset holding. The paper is organized as follows. The next section presents a model of an asset market with two types of investors: optimists who expect a high rate of return for the risky asset and pessimists who expect a low rate of retum for the risky asset. The dynamics of the asset price and wea1th distribution are derived and characterized. Section 3 studies the asymptotic distribution of the plOportions of wealth held by each agent. Section 4 discusses the implicadons on conditional variance and relates our model 10 AROI models in econometrics. This section also discusses the model's relation 10 another statistical model with time varying parameters called STAR. Section 5 considers the investors who use slight1y more complex stratcgies. "Fundamentalists" behave similarly 10 optimists and pessimists in Section 2. They maximize the 6 expected utility given their beliefs about the process of the asset retums. "Contrarians" start with the assumption that the market on average is wrong, and invest a proportion of their wealth in the risky asset which is opposite to the average proportion of wealth investe<! in that asset. The price dynamics in a market with fundamentalists and contrarian s is examined. Finally, Section 6 presents our conclusions. 2. OPTIMISTS, PESSIMISTS AND THE ASSET PRICE We consider an asset martet where there are two types of investors. One type of investors called "optimists", hold higher expectation about the rate of return of the risky asset than the other type of investors called "pessimists." Besides this assumption of heterogeneous beliefs, our model is very much a standard model of asset pricing model used in macro/f"mance literature. Tune is continuous and investors have infinite borizons. 80th types of agents are assumed to bave identical preferences with constant relative risk aversion, so their instantaneous c utility function is U ( C) = -1--'C ,where 't > O, 't - 1, and where e stands for 1-'t consumption. There are two types of assets, a riskless asset with an instantaneous rate of return r and a risky asset whose price is P(t). Let D(O be the total dividends accumulated for a sbare of the risky asset from time O 10 t Instantaneous dividends are thus dD(t). Let WO(t) be the me wealth of the optimists, W'(t) the wealth of pessimists and W(O = WO(O + W'(t) is the total me wealth in economy. Let 11 ( t) be the proportion of wealth tbat an agent of type i invcsts in the risky asset and c'(t) the proportion consumed out of total income. Givcn the assumptions about the return of the assets, wealth evolves according to the following equation: 7 dW.1 (t) = ()..1 (t) dP( t) +dD( t) + (1-)..1 (t» rdt-c.1 (t) dt)W.1 (t) P(t) , (1) i= o, p The agents believe that the process followed by the retums of the risky asset is detennined by, dP(t) +dD(t} = aJ,dt + PdB(t} P( t} where i is equal to o for the optimists and p for the pessimists and a > al) . Thus, optimists o expect higher retums than pessimists for a given level of risle. We assume that the agents never change their beliefs. The influence of each belief on the assel price, however, changes as the wealth distribution between the two types of agent changes. It may be more realistic 10 assume that some agents change their beliefs ü their strategy yields much lower returns than the altemative one, but such an assumption will not change the qualitative results. Under this assumption, a successful strategy would increase the influence through two channels. The wealth of the agents thal use the strategy grows faster, and, al the same time, the strategy gains some new converts. Thus, the dynamics of the model would qualitatively look the same. V. ds} The objecu.e of tbe ageDIS is ID maximi.., B .-'·U[C(sll subject ID tbe budget constraint (l), and C(t) > O; W'(t) > O; W'(O) = Wo > O. Under these cooditions ud (a xl ' assuming p > ( 1 -'t ) [ 1-x) 2 + which always holds when the agent is more risk . 2a2-r 8 averse than an agent with logarithmic utility, the demand for the risky asset by each type of agent will be a constant proportion of their total wealth, as Menon (1969) shows. Merlon (1969) also shows that letting A,1 be the proportion of the risky asset in the portfolio of a type i agent A,1 = u1 - r ,and letting cl be the proportion of consumption on wealth, 't'JP l( [ = _ e 1 p - ( 1 -'t' ) (u 1 -r) 2 + r ]) . 't' 2a2't' If we let N(t) be the total number of shares outstanding at time t, market clearing implies, = A,°WO(t) + A,J'WJ'(t) P(t)N(t) , (2) WO( t) or if we let q( t) = -W~(t-:)~ = A,°q(t) + A,J'(l-q(t» N(t)P(t) (2') W(t) • The finn that issucd the sbares is assumed to have instantaneous profits of This will be the case ü, far example. dcmand is a fixed proportion of total wealth, the price of the good is constant and the average 0051, which fluctuates randomly over time. is the same for a11 levels of production at any given instant of time. The fum uses the profits to buy back its 9 shares and pay out dividends, so that = - dn· (t) dN( t) (P( t) +dP( t» + N( t) dD( t) (3) U sing the market equilibrium condition (2) and the relation between asset retums and the profits process given by (3), we can solve for the process of equilibrium asset retums. Then by using (1) and lto's formula, we can derive the stochastic process q(t), the proponíon of wealth held by the optimists1 This is done with the following proposition. • Proposition A. dP( t) +dD( t) = .--l°cOq( t) -lPcP(l-q( t) ) dt + rdt a) P(t) 1°(1-1°)q(t) +lP (1-1P) (l-q(t» + 1 odB(t). (5) 1°(1-1°) q( t) +lP (1-1P) (l-q( t» _CO-CP]dt _ q(t) (l-q(t» (l°-lP) (1°q(t)+lP(1-q(t»o2 dt l°-lP (1°(1-1°)q(t)+lP(1-1P) (1-q(t»)2 P + q( t) (l-q( t» (l°-l ) o dB( t) , l°(l-l°)q(t) +lP(l-lP) (l-q(t» Proof: See the Apppendix. The proposition shows that the equilibrium asset retums dcpend on random shocks lO the profits. More importantly, the proportion of optimists' wealth, q(t), affects the equilibrium asset