Share price formation, …nancial instability and accounting design Yuri Biondi Pierpaolo Giannoccoloy Preg CRG Working Paper, Ecole Polytechnique, Paris Last update 9 November 2009
Abstract In this paper we develop a model of share price formation driven by accounting and market structures. Heterogeneous investors are assumed to discover and process fundamental information disclosed by the accounting system. The information set available to share market investors is then jointly composed by market-driven and …rm-speci…c (non-market) information. From one side, the accounting system provides collective signals of fundamental information. From another side, the price system provides collective signals of market-driven information. Both structures are signi…cant for the formation of aggregate share market prices over time. We simulate share price formation under alternative accounting designs (namely, historical cost, fair value and target reverting accounting), and derive implications and recommendations for the concept and occurrence of speculative bubbles, the cyclical e¤ects of accounting design on share market evolution, and share market allocative e¢ ciencies.
JEL classi…cation: C63, D4, M41, M48, G1 Keywords: accounting information; asset pricing model; fundamentalism; chartism; large ‡uctuations; market microstructure, historical cost accounting, fair value accounting. Cnrs & Cnam, Preg CRG-UMR7176, Ecole Polytechnique,
[email protected] of Bologna. Department of Economics.
[email protected].
y University
1
Electronic copy available at: http://ssrn.com/abstract=1502912
1
Introduction
Accounting, economics, and …nance literature pays increased attention to the link between share market dynamics and alternative accounting structures. In particular, the discovery and processing of accounting information is expected to play a speci…c role in the making of individual expectations and related decisions, in‡uencing then the formation of aggregate market prices over time. Following the conceptual framework by the US Financial Accounting Standards Board (FASB, CON 2 - par 98), “accounting information cannot avoid a¤ecting behavior, nor should it,” for accounting does integrate modes of management, governance, and regulation. This implies that alternative accounting representations cannot be “neutral” with respect to the underlying socio-economic activities, i.e., they cannot rest “without in‡uence on human behavior” (ibidem). Accordingly, the quality of …rm-speci…c information provided by the accounting system under accounting standards (or principles) becomes as much important as share market e¢ ciency in processing such information set that is available to investors. This information set is jointly composed by marketdriven and …rm-speci…c information. In particular, the accounting system (and regulation) conveys a speci…c representation of business a¤airs that shapes the payments from the business …rm to shareholders (in‡uencing share investment pays-o¤ in this way), and communicates collective signals to investors interested in the Share Exchange. This concept of a dual information set expands upon that adopted by the semi-strong form of market e¢ ciency developed by Fama (1970), that Fama and French (1992) related to …rm-speci…c information driven by fundamental analysis. Fama (1970) distinguishes three forms of share market e¢ ciency depending on the composition of the information set integrated by investors. Regarding the weak-form, the information set includes only the history of market prices; in the semi-strong form, it re‡ects all publicly available information; and the strong form tests it against all existing information. Our approach delves into the publicly available information set to disentangle two distinctive subsets: one driven by market (essentially, the history of prices), another comprising the …rm-speci…c information made available to investors by other institutional devices external to the market itself. The …nancial framework is then featured by two distinctive institutional dimensions: the share market, which endogenously generates a collective information driven by the market dynamics (Phelps 1987; Kirman 1999) – that is, the series of clearing prices …xed by the Share Exchange through time; and a not-market dimension, which generates a collective information generated outside the market, driven by institutions that are complementary to the market and that facilitate its making (Frydman 1982; Sunder 2002; Biondi 2008). This institutional approach originates a di¤erent perspective on share market dynamics (Shubik 1993; Sunder 1997). According to Fama (1991: 1575-1576), the main obstacle to inferences about market e¢ ciency (that he relates to col2
Electronic copy available at: http://ssrn.com/abstract=1502912
lective or individual “rationality”) is the joint-hypothesis problem that makes “market e¢ ciency per se not testable”: “we can only test whether information is properly re‡ected in prices in the context of a pricing model that de…nes the meaning of ’properly’.” (ibidem). Then, market e¢ ciency is jointly tested with some equilibrium model that must de…ne the way in which a certain information subset should be re‡ected by asset-pricing. Fama provisionally suggested that “the market e¢ ciency literature should be judged on how it improves our ability to describe the time-series and cross-section behavior of security returns” (ibidem). Our approach maintains the focus on the ‡ow of aggregate market prices through time, but it assumes that this ‡ow is jointly in‡uenced by two distinct sources of information: one driven by the market, another generated by other institutional devices such as established accounting processes of reporting and disclosure. This approach is then concerned with di¤erent paths of aggregate market prices depending on alternative institutional con…gurations, which feature this dual informational structure. The following analysis is expected to “enrich our knowledge on the behavior of returns across securities and through time” (Fama 1991: 1577), and may be extended to the impact of these alternative con…gurations on creation and allocation of resources among market participants, risk-sharing and bearing, and other featuring aggregate phenomena resulting from market dynamics. Drawing upon this framework of analysis, we develop a theoretical model that captures the special accounting role in the share price formation over time. The accounting system is then supposed to constitute a lighthouse in the amazing dynamics of share market through hazard, judgement and interaction. The rest of the paper is organized as follows. The …rst part presents the theoretical model, which comprises collective signals based on fundamental …nancial analysis, the formation of individual expectations and decision-making, and the evolution of the market price by matching aggregate demand and supply of shares through time. The second part simulates the model under alternative accounting designs. It investigates the relative impact on market price series of an accounting system that replicates the information provided by the market (so-called fair value accounting model), instead of constituting an autonomous source of …rm-speci…c information (so-called historical cost accounting model). Theoretically informed implications and recommendations are then derived on the cyclical e¤ects of accounting information on share market dynamics (Boyer 2007; Rochet 2008), and the allocative e¢ ciencies of share price formation.
3
Part I
The model 2
The conceptual framework
Consider a share market where only the shares of the business …rm j are quoted. The total number of its shares s is normalized to one, and, for each period t, the Share Exchange generates a market price of every share pt . Furthermore, the business …rm distributes dividends per share Dt based on the …nancial performance and position of the business …rm at time t 1. No discount nor alternative investment option rate apply.1 Risk-neutral investors i interested in this market conjecture about the share price-return relationship drawing upon heterogeneous mindsets: Ei;t = Ei;t (pt ; Dt j
t)
8i; t.
In particular, the individual investor’s expectations (and strategies) depend on a composite information set t that draws upon a monetary (cash ‡ow) and an epistemic (informational) basis. Both bases generate incentive structures that relate to the institutional con…guration (regulation) of the share market. Concerning the monetary basis, each investor forms his own expectations on the dividend ‡ow and the equity premium on share market price. The individual investor’s …nancial return (pay-o¤) depends then on the market price he may obtain by selling his shares (or the market price he should pay for buying the …rm’s shares), and on the dividend ‡ow that is distributed by the business …rm and is established according to accounting institutions, among others. Concerning the epistemic basis, the individual investor’s decision-making deals with two information ‡ows provided by distinctive institutional structures. One ‡ow of information is generated by the Share Exchange and is subsumed by the formation of aggregate (collective) pricing through time. Another ‡ow comes from accounting and other regulatory institutions that are complementary to the market, and that facilitate the working of the share market itself (Frydman 1982). In presence of heterogeneous individual mindsets, the price system and the accounting system complement each other in driving the market price formation trough time. The general system (which is no longer an equilibrium)2 consists in and depends upon the coherence and universal di¤usion of relevant and reliable knowledge by means of both a price system and an accounting system publicly determined and announced. The current period in-between ex ante and ex post locates here among future time, submitted to individual guesses and intentions, 1 This is not less restrictive than the widespread hypothesis of a …xed discount rate on the whole time period of analysis. 2 Our analysis distinguishes system and equilibrium as distinctive concepts.
4
hopes and fears, and past time, an history of reporting that, in principle, may be partly public, consistent, and conventionally agreed (Shackle 1967). In this context, the …nancial reporting provided by the accounting system is assumed to be common knowledge (Sunder 2002) that delivers relevant and reliable signals on the …nancial performance and position generated by the business …rm over time. These collective signals will be consistently integrated by individual investor’s expectations (or guesses) that drive his strategies. Therefore, individual investors make their …nancial decisions taking account this composite information set provided by distinctive institutional devices. Each investor decides how to exploit the market-driven information, the …rm-speci…c information, and the revision of its own guesses over time. The model assumes two di¤erent sorts of heterogeneity. First, two main classes of investors exist: The class I (for "Inside") comprises investors who currently hold shares (shareholders), and the class O (for "Outside") comprises investors who rest "outside the market" (that is, they currently do not hold shares). Each class of investors is expected to form individual expectations in distinctive ways, for entering the share market by holding shares arguably changes the individual stake and position respective to the share investment return dynamics. Furthermore, each investor is supposed to pay relative attention to the fundamental signal disclosed by the accounting system. This individual attention is captured by the parameter 'i 0 8i. Some investors speculate on the evolution of the market price and then utilize only the information generated by the Share Exchange, whilst other investors comprehensively look also at fundamental information on the …nancial performance and position of the business …rm over time. The evolution of the share market price will endogenously determine the composition of the shareholding basis, that is, the relative presence of fundamentalists and speculators "inside the market". Drawing upon their own individual expectations, investors decide a focal price and deliver bids and asks to the marketplace. The market design establishes then the aggregate share price that results from the interaction between them. The Share Exchange is the institutional device that collectively generates the aggregate market price pt through time. Such price generation may depend on either an exogenous time series ("t ), or an endogenous response to aggregate demand lack or excess (dt ) - the latter resulting from the mismatching of heterogeneous individual expectations generated by heterogeneous investors through time: pt = p(pt 1 ; dt ; "t ) 8t. The market price dynamics fundamentally depends on the matching of individual expectations, which is expected to have greater impact than the exogenous time series that captures a residual shock. The latter shock "t may be designed as a stationary or a random walk ("t = t 1 + N (0; 1), respectively with j j < 1 or j j = 1), a white noise "t N (0; 2 ), or another noise having a skewed distribution capturing the asymmetric likelihood of greater shocks. Theoretically speaking, this shock denotes all the in‡uences on the market price evolution that are not included in individual expectations and decisions. In a
5
sense, it relates to the actual degree of stock market e¢ ciency in Fama’s terms. A further extension may introduce a quota of shareholders who must liquidate their position because of budget, liquidity, and credit access constraints, and other individual preferences independent from price expectations.
3
The position of our approach in the literature
Our conceptual framework relates to the economics and …nance literature that has been modeling …nancial markets populated with boundly rational, heterogeneous investors using rules-of-thumb strategies (Hommes 2005). In particular, we draw upon the market dynamics between chartism (speculation) and fundamentalism to explore the contribution of the accounting structure to price formation on these markets. The accounting design is then understood as a speci…c institutional structure that plays a role in the market dynamics characterized by heterogeneous individual behavior, in line with Chiarella and Iori (2002), Horst (2005), and Anufriev and Panchenko (2009) among others. Such accounting structure is supposed to provide fundamental information and incentives to individual investors which shall integrate that information (and react to those incentives) according to the di¤erent strategies that drive their bids and asks on the market exchange. This interaction among individuals and between individuals and structures leads the aggregate formation of market prices over time and implies conditions of bounded (or intended) rationality that common knowledge provided by the accounting structure cannot overcome. In this context, both individual heterogeneity and accounting design are at the origin of trade and of eventual market excess volatility, if movements in prices induced by trading are larger than movements in underlying fundamentals. Individual decision-making and related market price formation are confronted with hazard, ignorance, and interaction. No such a thing as perfect foresight exists in such system that introduces degrees of freedom, chaos and complexity in a way that no equilibrium model may cope with. The system then is no longer captured by a unique steady state path. In particular, whilst individuals apply heuristics and biases to making decisions in this puzzling world, alternative accounting structures shape the market price paths over time. Therefore, the whole market delivers di¤erent pricing according to the ways information on (and incentives from) fundamentals are designed for and exploited by market participants. In particular, expectations of fundamental value computed (or guessed) by individual investors are de…nitely di¤erent from the fundamental information and signal collectively provided by the accounting system. This implies that market pricing depends on the ways individual expectations and collective fundamentals have been formed over time. The following sections will discuss the construction of the fundamental signal by accounting design; the formation of individual expectations and decisionmaking; the aggregation of demand and supply through matching pricing; and the resulting market price evolution through time. 6
4
Disentangling collective signals based upon …rmspeci…c information
Investors form their individual expectations on the basis of a composite set of information. The …rst subset is generated by the Share Exchange through the "price system", whilst the second subset comes from the accounting institutional framework and is provided by the "accounting system" of the business …rm. In sum, the information set available to investors for share investment decisionmaking comprises two kinds of collective information: market-driven information subsumed by the history of market price formation; …rm-speci…c information subsumed by …nancial reporting over time. Whilst the following section treats the individual discovery and processing of market-driven information, this section disentangles the …rm-speci…c collective signal Ft ( ) that each investor i integrates in his own mindset through the weight 'i 0 8i. This means that the only constraint imposed to the individual interpretation of the fundamental signal is that every investor attributes the same sign to the signal. A further extention may consider a disturbed or misunderstood signal. Consider the subset of fundamental information Yt;j that is available on period t and speci…c to the business …rm j. This subset may provide a lighthouse through the share market dynamics by delivering relevant and reliable collective signal Ft ( ) that will aid individual investors to form their own expectations: Ft = Ft ( jYt;j ) 8t.
(1)
Analytically, let assume that Ft ( ) is composed by three generic methods of evaluation, all based on the same set of information Yt;j . Consequently, individual investors may utilize one of these methods, or a linear combination of them: Ft = F1 ( jYt;j ) + F2 ( jYt;j ) + (1 with 0 ; 1.
) F3 ( jYt;j )
(2)
We de…ne each method according to empirically-based widespread methods of fundamental analysis: the …rst method is a ‡ow method that relates to price to earnings ratio analysis; the second method is a stock method that relates to the market to book ratio analysis; and the third method is an intelligence (or expert) method based on qualitative information on the speci…cs of the ongoing business …rm and its changing environment (Figure 1). On this basis: When Ft > 0, then the fundamental signal communicates positive perspectives, and individual investors expect that future prices may increase; 7
140
120
100
80
Median Capt ($100M) Price/Earnings (incl negative) Price/Book (*10)
60
40
20
0 31/03/78 31/07/80 30/11/82 29/03/85 31/07/87 30/11/89 31/03/92 29/07/94 29/11/96 31/03/99 31/07/01 30/11/03 31/01/77 31/05/79 30/09/81 31/01/84 30/05/86 30/09/88 31/01/91 28/05/93 29/09/95 30/01/98 31/05/00 30/09/02 31/01/05
Figure 1:
Median Market Capitalisation, Price to Earnings ratio, and Price to Book ratio
(S&P500; 1977-2005). Descriptive statistics based on S&P/Barra Monthly Indexes available at http://www.barra.com/Research/Description.aspx
When Ft < 0, then the fundamental signal communicates negative perspectives, and individual investors expect that future prices may decrease; When Ft = 0, then the fundamental signal conveys an unclear message, and individual investors do not derive any expectation from it. Theoretically speaking, this modeling of accounting information analysis to assess market pricing (Demsetz 1995: 93) is in line with unconstrained and constrained relationships between the market price series and available accounting information. In particular, our design expands upon Lev and Zarowin (1999) who, by de…ning pj;t = 0 + 1 F1j;t + 2 F2j;t + 3 F3j;t , analyze the unconstrained relationship between yearly average market prices pj;t , accounting measures of residual business income F1j;t and cumulated shareholders equity F2j;t , and other relevant information on the business …rm F3j;t (independent from F1j;t and F2j;t ). Accordingly,3 concerning the NYSE between 1977-1996, the yearly crosssectional association between share prices and accounting measures, as assessed by R2 , is above 0.9 during 1977 and 1988, and around 0.6 during 1989-1996 where R2 is here a measure of estimation error of p from accounting measures weighted by …rm, with weights obtained by cross-sectional annual regression. 3 See
also Nichols and Wahlen (2004).
8
Furthermore, our design corresponds with constrained relationships between the market price series and accounting information argued for by fundamental …nancial analysis literature (Dechow, Hutton, and Sloan 1999; Ohlson 1995; Feltham and Ohlson 1995; Ou and Penman 1989). This literature focuses on persistence of business incomes (determined by the accounting system), based either on time-series behavior or conditioning determinants (Lev and Thiagarajan 1993; Chant 1980; Freeman, Ohlson, and Penman 1982). In sum, heterogeneous investors may interpret the available fundamental information Yt;j to assess the gap between the current aggregate pricing pj;t and the fundamental pattern of the …nancial performance and position of the business …rm j. They may include this gap in their expectations in some extent, according to their views on the respective evolution of the business …rm and the market evaluation, and to their expected returns related to this evolution. Share price formation is then driven by two distinctive forces over time: the path of fundamental information, and the path of market pricing. In Lucas’s terms, this framework distinguishes indeed a "real" path (related to the accounting system and the business …rm) and a "monetary" path (related to the price system and the share market). Each path is generated by a distinctive institutional device and has speci…c cognitive features. In particular, every investor utilizes the real path to derive a personal assessment of the aggregate pricing, useful to adjust his expectations over time. The way the two paths interact through time essentially depends, therefore, on the formation of individual investor’s expectations and their revision through time.
5
The formation of individual investor’s expectations
This section analyzes how individual investors form their market price expectations. On the basis of these expectations, they will decide whether enter the Share Exchange by buying or selling, or simply wait (individual decisionmaking). The model assumes that two classes of investors exist: respectively shareholding and potential investors, that is, investors inside the market, and investors outside it. In addition, each investor is individually characterized by a certain degree of fundamentalism, that is, pays a relative attention to the fundamental signal Ft according to the individual parameter 'i . The relative proportion of each type among shareholding investors (inside the market) is endogenously determined by the share market dynamics. This means that the impact of di¤erent types of individual strategy depends on this dynamics (that in turn relates to the overall institutional con…guration), not only on subjective attitudes or beliefs. Every type of investors forms speci…c and timely expectations on the basis of the available information set comprising market-driven and …rm-speci…c in-
9
formation. The generic model4 for individual expectations Ei;t (pt+1 ) comprises the current price pt , the revision of past price expectation Ei;t 1 (pt ), the price trend t;t 1 (pt 1 ), and the composite signal from fundamental analysis Ft : Ei;t (pt+1 ) = Ei;t (pt ; Ei;t
1
(pt ) ;
t;t 1 (p); Ft )
8i; t:
In particular, each investors has de…nite preferences about the relative weights given to each component of the model: Investors pay a relative attention -according to the individual parameter 'i - to signals from fundamental analysis Ft based upon the …rm-specifc informational subset Yt;j . The latter includes history of dividends, accounting data, and qualitative information concerned with the overall performance and position generated by the business …rm through time. For instance, fundamentalists (with 'i > 0) may be considered as such investors that pay the cost of, or are capable to discover and process the …rm-speci…c information provided by the accounting system of the business …rm; Investors outside the market (which currently do not hold shares) do not revise their past price expectation, whilst investors inside the market (which currently hold shares) do.5 Analytically, let de…ne the following price expectation function which includes all the possible mindsets of the generic investor i: Ei;t (pt+1 ) = pt + 'i Ft ( ) + i (pt pt 1 ) + i (Ei;t 1 (pt ) pt ) 8i; t where i = 0 if i is outside the market, and 'i 0.
(3)
The individual heterogeneity in dealing with the price system (market pricing) is captured by the individual parameter i , and by the class parameter i . The …rst parameter represents the weight attributed to the market trend by individual decision-making (a degree of market con…dence), whilst the second parameter identi…es if the investor currently holds shares (and then revises his expectations on price over time), or not (and then he does not have previous expectations to be revised). Furthermore, F ( ) is the collective signal (see equations 1 and 2) that investors receive by fundamental analysis based upon the …rm-speci…c informational subset Yt . This signal concerns the change in the …nancial performance and position of the business …rm that investors may expect to be positively connected with changes in the share price over a long time horizon. The individual 4 This model of price expectation E St;I (pt+1 ) results from a combination between a "…rst order adaptive model": Et (Pt+1 ) = Et 1 (Pt ) + 0 (Pt Et 1 (Pt )) where 0 weights the revision of the most recent expectation error, and an "extrapolative expectation model": Et (Pt+1 ) (Pt ) = (Pt Pt 1 ) where weights the most recent price change (trend). Whenever > 0, any market price increase results in increasing the price expectation. 5 This is equivalent to set in equation (3) the corresponding parameter i = 0, which implies an "extrapolative expectation model", since potential speculators do not hold shares and consequently do not have past expectations on past market prices.
10
parameter 'i represents the degree of con…dence to the signal F ( ). In this article we analyze either the case in which 'i is equal to 0 - pure speculators or 1 - pure fundamentalists -, or the case in which 'i 2 (0; 1). A further extension may analyze the changing pattern of this price expectation, by allowing the revision of individual parameters (including ') according to individual learning over time (Frydman and Goldberg 2008).
6
Individual investor’s decision-making and strategy
This section analyzes how each type of investors decides his buy-or-sell strategy according to its expectations on share market price. Under conditions of radical heterogeneity and interaction between investors, and the dual informational (and institutional) structure, we maintain the basic rational expectation hypothesis that links individual strategies to the expected return (pays-o¤) R from the share investment: R = Ei;t (Gi;t
pt ) 8i; t
(4)
where Gi;t is the prospective potential gain. Accordingly investors i choose between returns today and future returns including capital gains, that is a form of Bellman’s equation: Gi;t = max arg fEi;t (pt+1 ) + Ei;t (Dt ) ; Gt+1 g .
(5)
This individual return relates then to the expected dividend ‡ow, the expected equity premium on share price period by period, and the prospective potential gain on future periods. These payo¤s are in‡uenced by the dual institutional con…guration of the share market. In particular, the dividend ‡ow fundamentally depends on the accounting system, whilst the equity premium fundamentally depends on the price system. The prospective potential gain combines options from both. Note that, at time t, the dividend ‡ow Dt and the related actual earnings (that constitute the upper bound of the dividend ‡ow) are not known. For sake of simplicity, let assume that all investors form their dividend expectation as follows: Ei;t (Dt ) = Dt 1 + aD (Dt 1 Dt 2 ) 8i. (6) The dividend ‡ow Dt may be su¢ ciently stable to justify this common expectation model based on dividend trend. Modeling individual expectations may be avoided by assuming that investors receive this information by common expert system or standardized procedure, instead of building themselves their own expectations on Dt . As every contracted or regulated business ‡ow (especially taxation), the dividend ‡ow ultimately relates to collective …rm-speci…c information Yt . A model re…nement may then allow fundamentalists to exploit 11
signals of fundamental analysis Ft (based upon Yt ) in order to better estimate the prospective dividend ‡ow generated by the business …rm over time. They may then look after earnings instead of dividends (Campbell and Shiller 1988). At each period t, investors decide their strategy according to their time horizon, by looking at the market side and the …rm-speci…c (non-market) side of the Share Exchange. Fundamental analysis allows them to analyze prospective expectations beyond the next period t + 1. They purport to derive all relevant and reliable information from signals of fundamental analysis based upon …rm-speci…c information. Therefore, they compare the current price pt to such prospective expectations on business performances that (are expected to) eventually lead the future price path in a longer horizon. However, if they decide to ignore the signal from fundamental analysis Ft (that is, 'i = 0), they will focalize only on two periods by taking into account time t and time t + 1. They compare then the current price pt to expectations on the dividend ‡ow Dt (which will be paid out during time t + 1) and the future share price pt+1 . Shareholding (potential) investors decide if sell (buy) or hold (wait to buy) shares as follows if pt Ek;t (Gt ) then investors ik;t wish to sell (wait to buy) if pt < Ek;t (Gt ) then investors ik;t hold (wish to buy) where k = I; O.
(7)
By assuming prospective expectations formed by shareholding (potential) investors, the equation (7) shows their best strategy. Note that no investor i is forced to sell immediately according to the results of his fundamental analysis. For instance, even though they expect, on a longer horizon, that the market price will eventually decrease according to the fall of prospective gain Gt from the business …rm j, they may nevertheless decide to hold shares if they believe that the next period price will be so high to deliver a convenient expected gain. In this case, fundamentalists act like short-termist speculators in order to maximize their expected return from share investment. Furthermore, at this stage, these individual strategies are only wishes, since they may or not meet a willing counterpart to eventually perform a share exchange. This matching step would realize according to the aggregation of individual demands and supplies on the marketplace. Given radical heterogeneity between interacting investors, speculative capital gains (or losses) are then made possible by inconsistent plans between them (Tirole 1982). The aggregate matching will be discussed in the next section.
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7
The formation of aggregate share market price over time
In this section we analyze the formation of aggregate share market price in order to simulate numerically the formation of aggregate share price under alternative accounting designs. The speci…ed model maintains the main theoretical assumptions discussed above. The number of investors is now normalized to one and a biunivocal correspondence exists - period by period - between each investor i and the parameter 'i 2 (0; 1), both inside the market (among shareholders) and outside (potential investors). For sake of simplicity, most of the individual heterogeneity is concentrated in this parameter. Then, (3) the generic expectation of an investor i becomes: Ei (pt+1 ) = pt +
(pt
pt
1)
+ "i jt + 'i (F )
(8)
pt
(9)
where "i jt
Ei (pt )
and =
0 if investors do not hold shares (potential demand) i if investors hold shares (potential supply).
In particular, contrary to the previous formulation (3), we assume that = . This concentrates the individual heterogeneity through the individual parameter 'i and i . The …rst parameter represents the degree of con…dence to the signal F ( ), whilst the second parameter identi…es if the investor currently holds shares (and then revises his expectations on price over time), or not (and then he does not have previous expectations to be revised). The pure fundamentalist and the pure speculator are then included as special cases with 'i = 0; 1. i
7.0.1
Individual mindsets
For sake of simplicity, let assume that there are not dividends. This assumption does not basically change the framework because we have assumed that all the investors have the same expectation on the prospective dividend ‡ow. Consequently, the individual decision-making by the investor i is captured by the his focal prices de…ned in equation (8): Ei;k (pt+1 ) where i 2 (0; 1) and k = I; O. In particular, let de…ne P0 that is the focal price of the pure speculator outside the market: E0;O (pt+1 ) = pt +
13
(pt
pt
1)
P0
(10)
Since the error "i jt - de…ned by equation (9) - is the linear combination of "0 jt (the error of the pure speculator) and "1 jt (the error of the pure fundamentalist), we assume without loss of generality that the parameter i is a linear combination of parameters 0 and 1 as follows: "'i jt =
0
(1
'i ) "0 jt +
1 'i " 1
jt .
(11)
Therefore, the expectation of the generic investor i (equation 8), inside or outside the market (holding shares), can be rewritten respectively as: Ei (pt+1 ) = P0 + "'i jt + 'i (F ) ,and Ei (pt+1 ) = P0 + 'i (F ) .
(12)
In this way, investors are ranked according to their distinctive 'i 2 (0; 1), that is, their relative degree of con…dence to the signal F ( ). 7.0.2
Aggregate Demand and Supply
According the main theoretical assumptions discussed above, the aggregation of individual demand and supply depends then on the making of individual expectations and related …nancial decisions. In particular, distinctive focal prices (Pi;k = Ei;k (pt+1 )) exist that induce each investor to change its position and enter the Share Exchange. These focal prices relate to the individual parameter 'i in equation (3), which represents the degree of con…dence to the signal F ( ). In particular, the extreme values of 'i = 0; 1 de…ne four representative investors: two with 'i = 0, who are "pure speculators" (inside or outside the market: SI and SO ) and do not care of the signal F ( ); and two with 'i = 1, who are "pure fundamentalists" (inside or outside the market: FI and FO ) and attribute the highest con…dence to the signal F ( ) derived by fundamental analysis. The individual decision-making is then captured by these four focal prices: Pi;k = Ei;k (pt+1 ) where i = 0; 1 and k = I; O. Given the equations (11) and (12), the aggregate demand and supply depend on these four focal prices (shareholding/potential fundamentalist and shareholding/potential speculator), since the time t = 0 when the …rm j o¤ers its shares on the primary market. Remember that, by assumption, the focal price of one investor may di¤er from the focal price of another one. The conditions to trade follow then the aggregation of individual demands and supplies based on focal prices. Let de…ne Pk Pk
max arg (E0;k (pt+1 ) ; E1;k (pt+1 )) min arg (E0;k (pt+1 ) ; E1;k (pt+1 )) where k = O; I. 14
(13)
All together, these focal prices determine a "marketable area" (that could not exist) where share exchanges are whished by some shareholding and potential investors (Figure 2). Inside and outside the market, investors observe the aggregate share market price pt and the fundamental signal Ft ( ) of the business …rm j and, according to their own expectations on pt+1 , decide whether change their position through selling or buying, or simply wait for the next period.
PO
PO
PI
PI Clearing
Demand Side
Supply Side
Area
Marketable Area Figure 2 - Aggregate demand and supply when clearing is possible according to representative investors’focal prices
The market design will decide how exchanges will be eventually performed within the "clearing area" (Figure 2). The latter denotes the width of the share market pricing for the period t. By assuming a linear distribution of investors, the supply S is de…ned by the following system: 0 0 if pt P I S = @ pt+1 P I if P I < pt < P I (14) 1 if pt P I . Furthermore, the demand D is de…ned by the following system: 0 1 if pt P O D = @ P O pt+1 if P O < pt < P O 0 if pt P O .
(15)
The clearing price arises when the demand is equal to the supply (S = D): S = D =)
pt+1
PI = PO
pt+1 if max arg (P I ; P O ) < pt < min arg never otherwise.
P I; P O
(16) The …gures 3 and 4 show the matching between aggregate demand and supply. In particular, the …gure 3 shows the case when the clearing price exists,
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Demand / Supply PO
1
Supply
PO
Demand
0
PI
Clearing
PI
Area
Marketable Area
Figure 3 - Aggregate demand and supply when clearing is possible
whilst …gure 4 shows the case when it does not. In the latter case, no share exchanges occur. Demand / Supply PO
Supply
PO
1
Demand
PI
0
Marketable Area Of Demand
PI
Marketable Area Of Supply
Figure 4 - Aggregate demand and supply when clearing is impossible
By computing, the clearing price is: ( P I +P O if max arg (P I ; P O ) < pt < min arg 2 pt+1 = 0 otherwise.
P I; P O
(17)
The determination of the clearing price passes through the identi…cation of P I ; P I ; P O and P O . In particular, their ranking depends on the sign of the signal Ft ( ). For sake of simulation, when the current clearing price does not exist, the market maker is expected to intervene by announcing the last available market price plus a small random tick value 0:01 U (0; 1). Therefore, the Share Exchange is here the institutional device that, according to its own design, collectively generates the market price pt over time. In particular, each period, the market maker …lls all the orders posted by individual investors according to their focal prices in a limit order book that ranks 16
these orders according to their reference price. The orders are then satis…ed (if possible) according to this ranking. This mechanism of aggregating demand and supply is designed as a one period batch auction where investors simultaneously post buy or sell orders, and the market maker calls an aggregate price series in search of bettering aggregate clearing and matching of individual demands and supplies over time.6 Accordingly, the resulting evolution of the aggregate market price pt is p t = pt
"dD 1 + "dS with " > "t .
"t
(18)
This market microstructure implies that collective pricing does not simply result from spontaneous and always perfect matching of individual expectations, since the latter may di¤er each from another and also be disappointed in some cases. The market may then experience aggregate lack of demand or supply, and even lack of transactions, at the called price. In the latter case, the market maker will call again the price of the previous period with a (positive or negative) tick value: pt = pt 1 "t . In this way, the Share Exchange generates a time series of aggregate market prices. This series is jointly in‡uenced by two distinct forces: one driven by the price system generated by individual bids and asks, another driven by the accounting system that denotes established processes of reporting and disclosure of fundamental information about the business …rm. The double information set available to investors at time t includes then present and past fundamental informations and prices7 . This dual architecture applies to analyze di¤erent paths of market prices depending on alternative accounting designs. For this, the rest of the paper will specify the theoretical model and provide the simulation.
Part II
Simulation 8
Simulation analysis and implications
The model allows to investigate the theoretical conditions under which a excess ‡uctuation in share price (a "speculative bubble"8 ) may occur. A overvaluation bubble is likely to occur when the market price walk persistently disconnects from the fundamental signal through time on a growing path. In this case, fundamentalist shareholders would increasingly sell their shares (after an initial 6 The
presence of t may then result from the working of a "drunk auctioneer". speaking, investors cannot react to the contemporaneous equilibrium price, and they trade using market orders. 8 Bubbles cannot be de…ned here on the basis of a collective concept of fundamental value that no longer exists. They may be denoted according to the stability and resilience of share price formation over time. 7 Analytically
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time when they mimic speculators and hold the shares even if they believe that their shares are overvalued), whilst potential fundamentalists outside the market would not buy. This dynamics will modify the composition of both the shareholding basis and the participants to the Share Exchange. Only speculators will go on buying and selling the shares, following their mindset based on the price trend and a shorter time horizon. Accordingly, since speculators dominate the market, a reversal in the price trend will eventually lead to a massive selling of shares by (speculative) shareholders, involving a price fall. At that moment, the market may attract again fundamentalist buyers, who start considering the price as undervalued, following the Rockefeller’s adage: "the time to buy is when there is blood in the streets". This framework of analysis maintains the long-horizon connection between fundamentals and market share pricing, but identi…es two distinctive patterns of information (or signals) for both sides of the Share Exchange. This approach applies then to assess the relative impact on market price formation by alternative accounting designs. In particular, two main accounting systems exist: one based on fair value accounting, another on historical cost accounting (Anthony 2004). They respectively perform: Fair value Accounting (FVA) model: the collective provision of such …rmspeci…c information that follows as near as possible the information provided by the market (marked-to-market accounting);9 Historical Cost Accounting (HCA) model: the collective provision of …rmspeci…c accounting information that constitutes an autonomous source of fundamental information. In the …rst case, the model may be simply calibrated on a signal F ( ) that reproduces the change in the market price level with one period lag (that is, 1 U (0; 1).10 If clearing pt 1 pt 2 ) plus a small random error 2 (U (0; 1) prices pt 1 and pt 2 do not exist, a small random error (U (0; 1) U (0; 1)) appears. In the second case, the model may be calibrated on a signal F ( ) that constitutes an exogenous time series (U (0; 1) U (0; 1)), that is, a random error with a maximum width of magnitude ten times inferior to the initial price p0 = 10. In order to focalize the simulation results only on the relative impact of the accounting design, we perform all the simulations under the same set of parameters: = 0:5; 0 = 0:25; 1 = 0:5; "0 = 0:01; "1 = 0:02; Ft=0 (U (0; 1) U (0; 1). On this basis, we simulate the price series over the …rst 100 periods, and we replicate the simulation for 200 000 times, for each 9 We
then consider marked-to-market accounting and fair value accounting synonymously. While the …rst design implies the use of observable market price to measure the value of an asset or a liability, the second design includes the recourse to observable and non observable inputs to reproduce that value. 1 0 We tested this accounting design without such random error, which may be an estimate error or bias, and the results did not basically change.
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alternative accounting design.11 Following the Montecarlo method, we measure and compare respectively: the mean average price, the average median price, the average price range (de…ned as the di¤erence between the third and the …rst quartiles of the price series), and the average relative price range (de…ned as the price range scaled by the average price). The Tables 1 to 4 provide the simulation results: Table I Average Price SD MAX MIN Two Tails T-Test
HCA 9.85 2.14 19.71 2.00 200.2
FVA 26.63 37.42 340.37 1.19
