A Simple-Model of Capital-Market Equilibrium with Incomplete Information

The Journal of FinanceVolume 42, Issue 3 p. 483-510 ArticleFree Access A Simple Model of Capital Market Equilibrium with Incomplete Information ROBERT C. MERTON, ROBERT C. MERTONJ. C. Penney Professor of Management, A. P. Sloan School of Management, Massachusetts Institute of Technology. My thanks to F. Black, C. Huang, S. Myers, R. Ruback and M. Scholes for helpful comments. I am grateful to J. Meehan for computational assistance.Search for more papers by this author ROBERT C. MERTON, ROBERT C. MERTONJ. C. Penney Professor of Management, A. P. Sloan School of Management, Massachusetts Institute of Technology. My thanks to F. Black, C. Huang, S. Myers, R. Ruback and M. Scholes for helpful comments. I am grateful to J. Meehan for computational assistance.Search for more papers by this author First published: July 1987 https://doi.org/10.1111/j.1540-6261.1987.tb04565.xCitations: 2,059 AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat I. Prologue The sphere of model financial economics encompasses finance, micro investment theory and much of the economics of uncertainty. As is evident from its influence on other branches of economics including public finance, industrial organization and monetary theory, the boundaries of this sphere are both permeable and flexible. The complex interactions of time and uncertainty guarantee intellectual challenge and intrinsic excitement to the study of financial economics. Indeed, the mathematics of the subject contain some of the most interesting applications of probability and optimization theory. But for all its mathematical refinement, the research has nevertheless had a direct and significant influence on practice. It was not always thus. Thirty years ago, finance theory was little more than a collection of anecdotes, rules of thumb, and manipulations of accounting data with an almost exclusive focus on corporate financial management. There is no need in this meeting of the guild to recount the subsequent evolution from this conceptual potpourri to a rigorous economic theory subjected to systematic empirical examination.11 Elements of this evolution are described in Merton (1983), (1987), forthcoming). Nor is there a need on this occasion to document the wide-ranging impact of the research on finance practice.22 The impact of efficient market theory, portfolio selection, risk analysis and contingent-claim pricing theory on money management and capital budgeting procedures is evident from even a casual comparison of current practices with those of twenty years ago. Financial research has also influenced legal proceedings such as appraisal cases, rate of return hearings for regulated industries, takeover rules, and “prudent-person” laws governing behavior of fiduciaries. The role of finance theory in the current wave of financial innovations is well documented in numerous articles in the financial press. See also Van Horne (1985). I simply note that the conjoining of intrinsic intellectual interest with extrinsic application is a prevailing theme of research in financial economics. The later stages of this successful evolution have however been marked by a substantial accumulation of empirical anomalies; discoveries of theoretical inconsistencies; and a well-founded concern about the statistical power of many of the test methodologies.33 Cf., “Symposium on Some Anomalous Evidence on Capital Market Efficiency,” Journal of Financial Economics, June-September 1978; Black (1986), Summers (1986) and R. C. Merton (1987). Finance thus finds itself today in the seemingly-paradoxical position of having more questions and empirical puzzles than at the start of its moderna development.44 Black (1986) describes many of the puzzles and attributes them to noise. Indeed, he predicts that, “ … research will be seen as a process leading to reliable and relevant conclusions only very rarely, because of the noise that creeps in at every step.” (p. 530) To be sure, some of the empirical anomalies will eventually be shown to be mere statistical artifacts. However, just as surely, others will not be so easily dismissed. I see this new-found ignorance in finance as mostly of the useful type that reflects our “… express recognition of what is not yet known, but needs to be known in order to lay the foundation for still more knowledge.?”55 So writes the sociologist of science, R. K. Merton (1987), who calls this particular form of ignorance, “specified ignorance.” Noting its common occurrence across both time and scientific fields, he points out that” … as the history of thought, both great and small, attests, specified ignorance is often a first step toward supplanting that ignorance with knowledge.” Anomalous empirical evidence has indeed stimulated wide-ranging research efforts to make explicit the theoretical and empirical limitations of the basic finance model with its frictionless markets, complete information, and rational, optimizing economic behavior. Although much has been done, this research line is far from closure. Some hold that the paradigm of rational and optimal behavior must be largely discarded if knowledge in finance is to significantly advance. Others believe that most of the important empirical anomalies surrounding the current theory can be resolved within that traditional paradigm.66 For various views on this issue, see the papers in Hogarth and Reder (1986). From these papers, it is evident that finance, and especially, speculative markets, provides a potentially-rich “strategic research site” (see R. K. Merton, 1987) for psychologists and sociologists as well as economists. Whichever view emerges as the dominant theme in finance, our understanding of the subject promises to be greatly enriched by these research programs. Although I must confess to a traditional view on the central role of rational behavior in finance, I also believe that financial models based on frictionless markets and complete information are often inadequate to capture the complexity of rationality in action. For example, in the modern tradition of finance, financial economic organizations are regarded as existing primarily because of the functions they serve and are, therefore, endogenous to the theory. Yet, derived rational behavior in a perfect-market setting rarely provides explicit and important roles for either financial institutions, complicated financial instruments and contracts, or regulatory constraints, despite their observed abundance in the real financial world. Moreover, the time scale for adjustments in the structures of financial institutions, regulations and business practices is wholly different than the one for either adjustment of investor portfolios or changes in security prices. Thus, even if all such structural changes served to accommodate individuals' otherwise unconstrained optimal plans, current (and perhaps, suboptimal) institutional forms can significantly affect rational financial behavior for a considerable period of time. Consider, for instance, the perfect-market assumption that firms can instantaneously raise sufficient capital to take advantage of profitable investment opportunities. This specification may be adequate to derive the general properties of investment and financing behavior by business firms on a time scale of sufficiently-long duration. It is, however, almost surely too crude an abstraction for the study of the detailed microstructure of speculative markets. On the time scale of trading opportunities, the capital stock of dealers, market makers and traders is essentially fixed. Entry into the dealer business is neither costless nor instantaneous. Thus, margin and other regulatory capital requirements can place an effective constraint on the number of opportunities that these profoessionals may undertake at a given point in time. Hence, these institutional factors may cause the short-run marginal cost of capital for these financial firms to vary dramatically over short time intervals. Therefore, to abstract from these factors may be to neglect an order-one influence on the short-run behavior of security prices. Similarly, models that posit the usual tâtonnement process for equilibrium asset-price formation do not explicitly provide a functional role for the complicated and dynamic system of dealers, market makers and traders observed in the real world. It would, thus, be no surprise that such models generate limited insights into market activities and price formation on this time scale. The expressed recognition of a nontâtonnement process for speculative-price formation is probably only important in studies of very short-run behavior. The limitations of the perfect-market model are not however confined solely to such analyses. The acquisition of information and its dissemination to other economic units are, as we all know, central activities in all areas of finance, and especially so in capital markets. As we also know, asset pricing models typically assume both that the diffusion of every type of publicly available information takes place instantaneously among all investors and that investors act on the information as soon as it is received. Whether so simple an information structure is adequate to describe empirical asset-price behavior depends on both the nature of the information and the time scale of the analysis. It may, for example, be reasonable to expect rapid reactions in prices to the announcement through standard channels of new data (e.g., earnings or dividend announcements) that can be readily evaluated by investors using generally-accepted structural models. Consider, however, the informational event of publication in a scientific journal of the empirical discovery of an anomalous profit opportunity (e.g., smaller-capital-ized firms earn excessive risk-adjusted average returns). The expected duration between the creation of this investment opportunity and its elimination by rational investor actions in the market plate can be considerable. Before results are published, an anomaly must in fact exist for a long enough period of time to permit sufficient statistical documentation.” After publication, the diffusion rate of this type of information from this source is likely to be significantly slower than for an earnings announcement. If the anomaly applies to a large collection of securities (e.g., all small stocks), then its “correction” will require the actions of many investors. If an investor does not know about the anomaly, he will not, of course, act to correct it. Once an investor becomes aware of a study, he must decide whether the reported historical relations will apply in the future. On the expected duration of this decision, I need only mention that six years have passed since publication of the first study on the “small-firm effect”, and we in academic finance have yet to agree on whether it even exists. Resolving this issue is presumably no easier a task for investors. Beyond this decision, the investor must also determine whether the potential gains to him are sufficient to warrant the cost of implementing the strategy. Included in the cost are the time and expense required to build the model and create the data base necessary to support the strategy. Moreover, professional money managers may have to expend further time and resources to market the strategy to clients and to satisfy prudence requirements before implementation. If profitable implementation requires regulatory and business practice changes or the creation of either new markets or new channels of intermediation, then the delay between announcement of an anomaly and its elimination by corrective action in the market place can, indeed, be a long one. Much the same story applies in varying degrees to the adoption in practice of new structural models of evaluation (e.g., option pricing models) and to the diffusion of innovations in financial products (cf. Rogers, 1972 for a general discussion of the diffusion of innovations). Recognition of the different speeds of information diffusion is particularly important in empirical research where the growth in sophisticated and sensitive techniques to test evermore-refined financial-behavioral patterns severely strains the simple information structure of our asset pricing models. To avoid inadvertent positing of a “Connecticut Yankee in King Arthur's Court,” empirical studies that use long historical time series to test financial-market hypotheses should take care to account for the evolution of institutions and information technologies during the sample period. It is, for example, common in tests of the weak form of the Efficient Market Hypothesis to assume that real-world investors at the time of their portfolio decisions had access to the complete prior history of all stock returns. When, however, investors' decisions were made, the price data may not have been in reasonably-accessible form and the computational technology necessary to analyze all these data may not even have been invented. In such cases, the classification of all prior price data as part of the publicly available information set may introduce an important bias against the null hypothesis. All of this is not to say that the perfect-market model has not been and will not continue to be a useful abstraction for financial analysis. The model may indeed provide the best description of the financial system in the long run.88 In tests of market efficiency, it is typically assumed that actual investors in real time knew (or should have known) the model and the statistical results derived by the researcher. As is well known, the large volatility of stock prices often requires a very long time series before a statistically-significant, estimated mean can be derived. Perhaps such studies should report the length of time before the discovered anomaly exceeded generally-accepted confidence intervals. It does, however, suggest that researchers be cognizant of the insensitivity of this model to institutional complexities and explicitly assess the limits of precision that can be reasonably expected from its predictions about the nature and timing of financial behavior. Moreover, I believe that even modest recognition of institutional structures and information costs can go a long way toward explaining financial behavior that is otherwise seen as anomalous to the standard friction-less-market model. To illustrate this thesis, I now turn to the development of a simple model of capital market equilibrium with incomplete information. II. Capital Market Equilibrium With Incomplete Information In this section, we develop a two-period model of capital market equilibrium in an environment where each investor knows only about a subset of the available securities. In subsequent sections, we explore the impact on the structure of equilibrium asset prices caused by this particular type of incomplete information. There are n firms in the economy whose end-of-period cash flows are technologically specified by: C ~ k = I k [ μ k + a k Y ~ + s k ϵ ~ k ] (1) where Y ~ denotes a random variable common factor with E ( Y ~ ) = 0 and E ( Y ~ 2 ) = 1 and E ( ϵ ~ k ) = E ( ϵ ~ k | ϵ 1 , ϵ 2 , … , ϵ k − 1 , ϵ k + 1 , … , ϵ n , Y ) = 0 , k = 1 , … , n . I k denotes the amount of physical investment in firm k and μ k , a k , and S k represent parameters of firm k's production technology. Let V k denote the equilibrium value of firm k at the beginning of the period. If R ~ k is the equilibrium return per dollar from investing in the firm over the period, then R ~ k ≡ C ~ k / V k , and R ~ k = R ¯ k + b k Y ~ + σ k ϵ ~ k , (2) where from (1), R ¯ k = E ( R ~ k ) = I k μ k / V k ; b k = a k I k / V k and σ k = s k I k / V k , k = 1 , … , n . By inspection of (2), the structure of returns is like that of the Sharpe (1964) “diagonal” model or the “one-factor” version of the Ross (1976) Arbitrage Pricing Theory (APT) model. In addition to shares in the firms, there are two other traded securities: a riskless security with sure return per dollar R and a security that combines the riskless security and a forward contract with cash settlement on the observed factor index Y. Without loss of generality, it is assumed that the forward price of the contract is such that the standard deviation of the equilibrium return on the security is unity. Thus, the return on the security can be written as: R ~ n + 1 = R ¯ n + 1 + Y ~ . (3) It is assumed that both this and the riskless security are “inside” securities and therefore, investors' aggregate demand for each of them must be zero in equilibrium. The model assumes the standard frictionless-market conditions of no taxes, no transactions costs, and borrowing and shortselling without restriction. There are N investors where N is sufficiently large and the disshstribution of wealth sufficiently disperse that each acts as a price taker. Each investor is risk averse and selects an optimal portfolio according to the Markowitz-Tobin mean-variance criterion applied to end-of-period wealth. The preference of investor j is represented as: U j = E ( R ~ j W j ) − δ j 2 W j Var ( R ~ j W j ) , (4) where W j denotes the value of his initial endowment of shares in the firms evaluated at equilibrium prices; R ~ j denotes the return per dollar on his portfolio; and δ j > 0 , j = 1 , … , N . In addition to an initial endowment of shares, each investor is endowed with an information set described as follows: Common knowledge in all investors' information sets includes the return on the riskless security; the expected return and variance of the forward contract security given in (3); and the basic structure of securities' return given in (2). However, for any given security k, knowledge of the specific parameter values in (2) may not be included in some investors' information sets. An investor is said to be “informed (know) about security k” if he knows ( R ¯ k , b k , σ k 2 ) . All investors who know about security k agree on these parameters values (i.e., conditional homogeneous beliefs). Let J j denote a collection of integers such that k is an element of J j if investor j knows about security k , k = 1 , … , n + 2 , where security n + 2 is the riskless security. Thus, by assumption, n + 1 and n + 2 are contained in J j , j = 1 , … , N . If J j contains all the integers k = 1 , … , n + 2 , then investor j's information set is complete. Although the model does not rule out this possibility for some investors, if all investors' information sets were complete, then the model would reduce to the standard Sharpe-Lintner-Mossin Capital Asset Pricing Model. Therefore, it is assumed that investors generally know only about a subset of the available securities and that these subsets differ across investors. The key behavioral assumption of the model is that an investor uses security k in constructing his optimal portfolio only if the investor knows about security k. The prime motivation for this assumption is the plain fact that the portfolios held by actual investors (both individual and institutional) contain only a small fraction of the thousands of traded securities available.99 Cf. Friend and Blume (1975) and Blume and Friend (1978),(1986). There are, of course, a number of other factors (e.g., market segmentation and institutional restrictions including limitations on short sales, taxes, transactions costs, liquidity, imperfect divisibility of securities) in addition to incomplete information that in varying degrees, could contribute to this observed behavior.1010 Cf. Amihud and Mendelson (1986); Brennan (1970); Constantinides (1984); Errunza and Losq (1985); Levy (1978); and Miller (1977). Because this behavior can be derived from a variety of underlying structural assumptions, the formally-derived equilibrium-pricing results are the theoretical analog to reduced-form equations. As in the Grossman-Stiglitz (1976) single-security model of asymmetric-information trading, the conditional-homogeneous-beliefs assumption posited here ensures that all informed traders in security k have the same information about security k. However, in contrast to their analysis, the issues of gaming between informed and uninformed investors that surround trading in an asymmetric information environment do not arise here because only equally-informed investors trade in each security. Concern about asymmetric information among investors could be an important reason why some institutional and individual investors do not invest at all in certain securities, such as shares in relatively small firms with few stockholders. However, as is evident from the Grossman-Stiglitz analysis, such concerns about informed traders are not alone adequate to render this polar extreme in behavior as optimal. Therefore, I discuss briefly other types of information cost structures that could lead to the posited behavior in our model. For this purpose, it is useful to think of information costs as partitioned into two parts: (1) the cost of gathering and processing data and (2) the cost of transmitting information from one party to another. A prime source of information about a particular firm is, of course, the firm itself. Information required by investors overlaps considerably with the information managers require to operate the firm. Hence, the firm's marginal cost for gathering and processing the data needed by investors would seem to be small. Nevertheless, as is evident from the extensive literature on the principal-agent problem and signalling models,1111 See Bhattacharya (forthcoming) for an extensive review. the cost of transmitting this information to investors so that they will use it efficiently, can be considerable. The signalling models are focused on the problem of the firm transmitting to investors specific information such as earnings prospects and investment plans. The types of costs emphasized are the incentive costs necessary to induce managers to transmit information and the costs required to make credible the information they transmit. It is generally assumed in these models that all public (“non-insider”) investors receive the same information whether they are currently shareholders or not. In the Bawa-Klein-Barry-Brown models of differential information in which each investor has the same information set,1212 Klein and Bawa (1977); Barry and Brown (1984), (1985), (1986). the focus of analysis is on the price effects from differences in the quality of information across securities (i.e., parameter-estimation risks). In contrast, our model assumes that the quality of information (i.e., the precision of the estimates of R ¯ k , b k , σ k ) is the same for all securities, and focuses on the price effects from different distributions of that information across investors. Thus, the differential-information models cover the price effects of differences in the depth of investor cognizance among securities, whereas the emphasis here is on the differences in the breadth of investor cognizance. Although the types of costs underlying the signalling and differential information models would surely be an important part of a more-detailed information-cost structure for our model, there is another type of cost that logically proceeds them: namely, the cost of making investors aware of the firm. That is, for Party A to convey useful information to Party B, requires not only that Party A has a transmitter and sends an accurate message, but also that Party B has a receiver. If an investor does not follow a particular firm, then an earnings or other specific announcement about that firm is not likely to cause that investor to take a position in the firm.1313 Although larger volatility and lower substitutability among equities will surely make the derived effects greater for equity securities than debt, the same idea applies to all securities. For example, a bond trader who responds quickly to interest rate news by trading U.S. Treasury bonds, may not be willing to trade GNMA mortgage-backed bonds unless he has borne the set-up costs necessary to understand the effect on price of the prepayment feature of these bonds. If, for each firm, investors must pay a significant “set-up” (or “receiver”) cost before they can process detailed information released from time to time about the firm, then this fixed cost will cause any one investor to follow only a subset of the traded securities. Because this fixed cost is a “sunk cost” for existing shareholders, the effective information received by current shareholders, even from a public announcement by the firm, will not be the same as that received by other investors. The firm is of course, not the only source of information available to investors. There are stock market advisory services, brokerage houses, and professional portfolio managers. However, much the same argument used here for the firm can also be applied to the costs in making investors aware of these sources.1414 As with the firm, information from brokerage or investment services will only influence an investor's decisions if he knows about the source and has incurred the set-up cost to properly calibrate the information. Similarly, the investor knows about only a small number of money management institutions and if he does not know about an institution, he will not invest with it. Our background information-cost story fits well with the Arbel-Carvell-Strebel theory of “generic” or “neglected” stocks.1515 See Arbel and Strebel (1982); Arbel, Carvell and Strebel (1983); and especially, Arbel (1985, p. 5); and Strebel and Carvell (1987, Chapter 1). In their theory, neglected stocks are ones that are not followed by large numbers of professional analysts on a regular basis. They assume that if the quantity of analysts following a stock is relatively small, then the quality of information available on the stock is relatively low. From this, they conclude that ceteris-paribus, equilibrium expected returns on neglected stocks will be larger than on widely-followed stocks. Although our simple model posits no differences in the quality of information across securities, it is clear in our model that if the number of investors that know about security k is relatively small, then security k would fit the definition of a neglected security in the Arbel-Carvell-Strebel model. With this, we close our discussion on the information cost structure underlying the model. With the structure of the model established, we turn now to the solution of the portfolio selection problem for investor j. If w k j denotes the fraction of initial wealth allocated to security k by investor j, then from (2) and (3), the return on the portfolio can be written as: R ~ j = R ¯ j + b j Y ~ + σ j ϵ ~ j (5) where: b j ≡ ∑ 1 n w k j b k + w n + 1 j σ j ≡ ∑ 1 n ( w k j ) 2 σ k 2 ϵ ~ j ≡ ∑ 1 n w k j σ k ϵ ~ k / σ j . From (2), we have that E ( ϵ j | Y ) = E ( ϵ j ) = 0 . Using the condition that w n + 2 j = 1 − ∑ 1 n + 1 w k j and substituting b j − ∑ 1 n w k j b k for w n + 1 j , we can write the variance and expected return on the portfolio as: Var ( R ~ j ) = ( b j ) 2 + ∑ 1 n ( w k j ) 2 σ k 2 (6.a) and R ¯ j = R + b j ( R ¯ n + 1 − R ) + ∑ 1 n w k j Δ k (6.b) where Δ k ≡ R ¯ k − R − b k ( R ¯ n + 1 − R ) . From (4), the optimal portfolio choice for the investor can be formulated as the solution to the constrained maximization problem: Max { b j , w j } [ R ¯ j − δ j 2 Var ( R ~ j ) − ∑ 1 n λ k j w k j ] (7) where λ k j is the Kuhn-Tucker multiplier that reflects the constraint that investor j cannot invest in security k if he does not know about security k. That is, λ k j = 0 if k ∈ J j and w k j = 0 if k ∈ J j c , the compliment to J j , k = 1 , … , n . From (6.a) and (6.b), the first-order conditions for (7) can be written as: 0 = R ¯ n + 1 − R − δ j b j (8.a) 0 = Δ k − δ j w k j σ k 2 − λ k j , k = 1 , … , n . (8.b) From (8.a) and (8.b), the optimal common-factor exposure and portfolio weights can be written as: b j = [ R ¯ n + 1 − R ] / δ j (9.a) w k j = Δ k / ( δ j σ k 2 ) , k ∈ J j (9.b) w k j = 0 , k ∈ J j c (9.c) w n + 1 j = b j − ∑ 1 n w k j b k (9.d) w n + 2 j = 1 − b j − ∑ 1 n w k j ( b k − 1 ) (9.e) From (8.b) and (9.c), we have that λ k j = Δ k , k ∈ J j c (10) and λ k j = 0 for k ∈ J j . By inspection of (10), the “shadow cost” of not knowing about security k is the same for all investors.1616 The shadow cost is measured in units of expected return. Given the return structure (2) and (3), equation (10) for the shadow cost applies not only for the mean-variance criterion, but for all concave utility maximizers. Having solved fo