A SIMPLE MODEL OF CAP ITAL MARKET EQUILIBRIUM WITH INCOMPLETE INFORMATIONT Working Paper #1869-87 March 1987 Presidential Address, American Finance Associatio New Orleans, December 29, 1986. Forthcoming, in J (July 1987
A SIMPLE MODEL OF CAPITAL MARKET EQUILIBRIUM WITH INCOMPLETE INFORMATIONt Robert C. Merton Working Paper #1869-87 March 1987 tPresidential Address, American Finance Association, New Orleans, December 29, 1986. Forthcoming, in Journal of Finance (July 1987)
A SIMPLE MODEL OF CAPITAL MARKET EQUILIBRIUM WITH INCOMPLETE INFORMATIO. Robert C. Merton x The sphere of modern financial economics encompases 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 Nor is there a need
A SIMPLE MODEL OF CAPITAL MARKET EQUILIBRIUM WITH INCOMPLETE INFORMATION Robert C. Merton * I. Prologue The sphere of modern financial economics encompases 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.1 Nor is there a need
on this occasion to document the wide-ranging impact of the research on finance practice. 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 has however been marked by a substantial accumulation of empirical anomalies; discoveries of theoretical inconsistencies; and a well-founded concern about the statistical power of 3 many of the test methodologies, Finance, thus finds itself today in the seemingly-paradoxical position of having more questions and empirical puzzles than at the start of its modern development k 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 need s to be known in order to lay the foundation for still more knowledge, ?3 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. Whichever view emerges as the dominant theme in finance, our understanding of the subject promises to be greatly enriched by these research programs
III -2- on this occasion to document the wide-ranging impact of the research on 2 finance practice. 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 has 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.3 Finance, thus finds itself today in the seemingly-paradoxical position of having more questions and empirical puzzles than at the start of its modern development. 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." 5 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. 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 ad justments in the structures of financial institutions, regulations and business practices is wholly different than the one for either ad justment 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 instantly 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
-3- 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 instantly 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 professionals 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 tatonnement 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 nontatonnement 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 lace 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 anal. It may, for example, be reasonable to expect rapid reactions in prices to the announcement through standard channels of new data (e.g
-4- regulatory capital requirements can place an effective constraint on the number of opportunities that these professionals 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 tatonnement 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 nontatonnement process for speculativeprice 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 -,Mace 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-capitalized firms earn excessive risk-ad justed average returns). The expected duration between the creation of this investment opportunity and its elimination by rational investor actions in the market place 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. 7 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
-5- 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-capitalized 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 place 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 technique s 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
III -6- 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 financialmarket 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. 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 frictionless-market model. To illustrate this thesis, I now turn to the development of a simple model of capital market equilibrium with incomplete information
-7- 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.8 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 frictionless-market model. To illustrate this thesis, I now turn to the development of a simple model of capital market equilibrium with incomplete information
8 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 where denotes a random variable common factor with E(Y)=0 E(Y)-1andE()-E(1,2…;-1,+1……,,Y)=0,k=1,…n I denotes the amount of physical investment in firm k and and s, represent parameters of firm k's production technology, Let v, denote the equilibrium value of firm k at the beginning of the period. If is the equilibrium return per dollar from investing in the firm over the period, then 段瓦1++k, (2) where from(1),x-e(r )=Ikk/k,"T and "%Kkk k - l,..., 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
-8- 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: ik -LIk Elk y +k (1) where Y denotes a random variable common factor with E(Y) = 0 and E(r) = 1 and E(Ek) = E(Ek l 1 C ,... 2 ,+ , l ,.. , n Y) = O , k =n. Ik denotes the amount of physical investment in firm k and 1 k' ak and sk represent parameters of firm k's production technology. Let Vk denote the equilibrium value of firm k at the beginning of the period. If Rk is the equilibrium return per dollar from investing in the firm over the period, then Rk = Ck/Vk, and % - % % R = + bkY + ki (2) R 2=R~+b~a , (2) where from (1), R = E(Rk) = Ikpk/Vk; bk = akk/Vk and = SkIk/Vk k = l,...,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+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 distribution 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 is represented as U,=E(RW> Var(R-N> (4) where denotes the value of his initial endowment of shares in the firms evaluated at equilibrium prices; Ry R denotes the return per dollar on his portfolio; and 5,>0,1=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
-9- 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 + '(3 RR Rn + Y *(3) n+l n+l 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 distribution 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: Uj = E(R W) Wj Var(RWA) ,(4) 2WJ where W denotes the value of his initial endowment of shares in the firms evaluated at equilibrium prices; R denotes the return per dollar on his portfolio; and 6> , j = .. .N. In addition to an initial endowment of shares, each investor is endowed with an information set described as follows: Common knowledge in all
