10 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'returns given in (2). However, for any given security k, knowledge of the specific parameter values in(2)may not included in some investors' information sets An investor is said to be"informed (know)about security k if he knows kk,,). All investors who know about security k agree on these parameter values(. e, conditional homogeneous beliefs).Let denote a collection of integers such that k is an element of J, if investor j knows about security k, k=l,..., n+ 2, where security n+ 2 is the riskless security. Thus, by assumption, n+1 and n+2are contained in J, j=1,., N. IfJ contains all the integers k= l,..., n+2, then investor J's information set is complete. Althoug h 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 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. 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
-10- 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' returns 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 (Rk,bk,ao). All investors who know about security k agree on these parameter values (i.e., conditional homogeneous beliefs). Let Jj denote a collection of integers such that k is an element of Jj 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 Jj, j = 1,...,N. If Jj 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.9 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 HIt
11 divisibility of securities) in addition to incomplete information that in varying degrees, could contribute to this observed behavior 10 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 (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
-11- divisibility of securities) in addition to incomplete information that in 10 varying degrees, could contribute to this observed behavior. 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, 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 ach investor has the same information set, 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 assume s that the quality of information (i.e, the precision of the estimates of R,,,,o, 1s 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 proceed s them: namely, the cost of making investors aware of the firm. That is, for Party a to convey useful information to Party B, requires
III -12- be small. Nevertheless, as is evident from the extensive literature on the principal-agent problem and signalling models, 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,l 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 Rk,bk,Ok) 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
13 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 irm, then an earnings or other specific announcement about that firm is not likely to cause that investor to take a position in the firu, 3 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 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 source Our background information-cost story fits well with the Arbel-Carvel Strebel theory of" generic or"neglected stocks. 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 abou
-13- 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.13 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 14 these sources. Our background information-cost story fits well with the Arbel-CarvellStrebel theory of "generic" or "neglected" stocks.1 5 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
14 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 1. If wi denotes the fraction of initial wealth allocated to security k by investor 1, then from (2)and (3), the return on the portfolio can be written as RJ=討+lY+dj From(2), we have that E(EIY)=E(E)=0.Using the condition that n+1,j and substituting b-∑ for w k n+1,we can write the variance and expected return on the portfolio as: Var(r)=( (6,a) bCR n+1 From(4), the optimal portfolio choice for the investor can be formulated as the solution to the constrained maximization problem:
-14- 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 . If wj denotes the fraction of initial wealth allocated to security k by investor , then from (2) and (3), the return on the portfolio can be written as: j = + b + aJL (5) where: b_ E wnJb + 1 kbk W + n+l E / E n (wJ)2 2 1 k k From (2), we have that E( j l IY) = E(Ej) = 0. Using the condition that wn+l ' b n+2 =1 E1 wk and substituting b j n+2 1 k n- for w we can I k k n+l write the variance and expected return on the portfolio as: Var(Rj) = (bJ) 2 + n (wj) 2 a2 (6.a) 1 k k and R R + b (R - R) + w (6.b) n+l 1 k ( where (R R) k Rk -R - bk(Rn+l - R) From (4), the optimal portfolio choice for the investor can be formulated as the solution to the constrained maximization problem: