[A] major problem of finance theory is an overreliance on the assump­tion of common beliefs.  I have a suspicion, (and it is only a suspi­cion), that this assumption is behind the failure of theory to explain much of the observed empirical behavior in markets.  It is difficult, however, to figure out how to proceed in relaxing this assumption.

                                      Richard Roll, 1988

5.1.  Introduction[1]

Most asset pricing theories assume investors hold common or homoge­neous beliefs.  These assumptions permit theorists to aggregate individual valuation formulas into a market-level expression.  Recently, common knowledge belief assumptions have begun replacing the older homogeneous beliefs assumptions.  If beliefs are common knowledge, then everyone knows everyone knows .... everyone's beliefs.  Without common knowledge assump­tions, investors in these models face intrinsic uncertainty over which firms the rest of the market regarded as undervalued or overvalued.  Investors might also face uncertainty over how the market's beliefs will change with the arrival of new information.  This uncertainty can affect future prices; consequently, it figures in any capital gains (loss) estimate.

    Stock prices largely reflect real earnings for the past year coupled with expected earnings for the out year or two.  Investor beliefs partially determine forecasts for future earnings and play an even larger role in the equilibrium price-to-earnings ratio for each firm's stock.  This chapter examines whether Modigliani-Miller (MM) arbitrage, the Capital Asset Pricing Model (CAPM), and Ross's Arbitrage Pricing Theory (APT) need to employ common knowledge investor belief assumptions.

    Although we could apply these concepts to any asset valuation theory, this chapter emphasizes the celebrated Modigliani-Miller (1958) paradigm.  The MM arbitrage strategy has a common thread to most of the other asset pricing theories.  The paradigm's thirty year anniversary in 1988 spawned several MM theory retrospectives, which surprisingly omitted any discussion of embedded common knowledge assumptions in the model.

    Concerning the informational structure of the paradigm, Miller (1988) writes `one of the key assumptions of both the leverage and dividend models (was) that all capital market participants, inside managers and outside investors alike, have the same information about the firm's cash flows.'  Once this assumption is relaxed, it is well known that the MM results do not hold under asymmetric information.  An extensive literature deals with this issue dating back to Ross (1977a) and Leland and Pyle (1977),[2]  who showed how managers with inside information might have incentives to `signal' the market about the firm's future earnings through either the firm's debt-equity ratio or dividend policy.  In general, this literature focuses on the related topic of managerial incentives but does not explore the theoretical underpinnings of the MM propositions.

    Recent developments on common knowledge raise questions about the robustness of the MM results as the model is currently formu­lated.  How are the MM results affected if investors agree (on the cash flows of a firm) without knowing they agree?[3]  Would some investors want to speculate that the market diverges from their own beliefs?  Stiglitz (1969, 1974) has argued that the MM paradigm remains valid even if heteroge­neous beliefs are admitted into the model.  However, his two-period (1969) model was inadequate to consider the problems posed by speculation, where traders plan to buy and resell knowing information will arrive before the state of the world is revealed.  His multiperiod (1974) model conveniently assumes that all the well-known aggregation  problems with diverse individual beliefs have somehow sorted themselves out, and the economy starts in a market-clearing equilibrium.

    Our analysis can be distinguished from the older, pre-common knowledge era literature on asymmetric information.  The older literature examined the impact of heterogeneous beliefs based on insider information, with managers or existing shareholders better informed than the rest of the market.  This chapter focuses on an MM world in which everyone is equally well informed.  Investors in our formulation will not know if their beliefs are homogeneous or heterogeneous.

    In studying the role of common knowledge in the MM paradigm, one finds a gap even in modern restatements of debt-equity irrelevancy.  Modern restatements are based on income ar­bitrage:  people switching from one asset to another to achieve a greater cash flow from earnings. In fact, they can also receive cash flows from capital gains.  For example, speculation and capital gains considerations do not enter into the following analysis.

        The values of debt and equity, in turn, are based on the cash flows that are expected to go to each.  Since the cash flow of the firm must go either to debt or to equity and since linearity assures us that the sum of their values is the value of the sum of their cash flows, it follows that the total value of the firm depends simply on the total cash flow to the firm, regardless of the debt-equity mix chosen by that firm.  Ross (1988), p.128.

Intelligent investors know the market values of firms depend on more than just accounting identities; market values depend on (changing) beliefs about the future.  We will assess the impact these speculative considerations have on MM-style arbitrage, the CAPM, and the APT.

    The CAPM is an equilibrium model, whereas the MM paradigm and the APT are arbitrage models.[4]  The absence of arbitrage opportunities permits market clearing in the latter two models.  The reader may be surprised to learn that we find the MM paradigm does not require investor beliefs to be common knowledge.  The CAPM implicitly makes beliefs common knowledge by endowing investors with perfect foresight about asset returns.  The APT needs common knowledge belief assumptions for its validity.  Thus the need for common knowledge belief assumptions depends on features of individual models, not on the equilibrium solution concept.

    Section 5.2 discusses MM's attempts to deal with intrinsic uncertainty and explains how it might lead to speculative invest­ments in the opposite direction of income arbitrage.  Section 5.3 illustrates the logical intuition in the original MM framework.  Section 5.4 presents a numerical example again based on the MM paradigm.  The necessity for common knowledge belief assumptions is examined for the Capital Asset Pricing Model in Section 5.5 and for Ross's Arbitrage Pricing Theory in Section 5.6.  Section 5.5 derives the CAPM from basic constructs; most finance textbooks merely state the CAPM without deriving it.  Section 5.7 examines whether the MM and APT pardigms can incorporate heterogeneous beliefs, as some have suggested, and remain valid.  Section 5.8 presents our conclusions and some observations about real world arbitrage.

5.2.  MM's Missing Assumption and Speculative Investment

MM-style arbitrage opportunities imply that when securities are either undervalued or overvalued, an investor should be able to buy and sell stocks and bonds in such a way that he increases the income he would have received from his original portfolio position.  In focusing on income, MM neglected the capital gains motive. 

    MM sensed their original proof of the debt-equity propositions left open the possibility for price speculation.  Although some investor personally regards a firm as overvalued, he may fear the rest of the market views that same firm as fair valued or even undervalued.  That concern led MM to include a missing `imputed rationality' assumption in their subsequent (1961) dividend paper:

        [W]e shall say that an individual trader `imputes rationality' to the market or satisfies the postulate of `imputed rationality' if, in forming expectations, he assumes that every other trader in the market is (a) rational in the previous sense of preferring more wealth to less regardless of the form an increment in wealth may take, and (b) imputes rationality to all other traders... Our postulate thus rules out, among other things, the possibility of speculative `bubbles'...  MM (1961).

    By imputed rationality, MM sought to eliminate self-fulfilling equilibria, in which investors take dividends into account simply because they believe others will do so.  MM obviously intended that subse­quent formulations of the capital structure paradigm assume imputed rationality as well.  Since prices are determined endogenously in capital markets, the beliefs of market participants will materially affect each person's own opportunity set, whether these beliefs hold that debt-equity ratios, dividends, or sunspots are relevant to firm valuations.   

    This missing assumption appears to have been noted by only one previous paper in the vast MM literature:

        MM clearly read much more into this assumption than a statement of the assumption seems to warrant....Thus by `rationality' MM apparently mean than an individual ignores capital gains aspects of security purchases - but this is never stated explicitly....And what this amounts to is an assump­tion that individuals behave in the market on the basis of earnings expectations (income prospects) and not on the basis of wealth considerations based on prospective future values of shares or bonds....[W]hen an obviously important (and perhaps dominant) feature or aspect of behavior is assumed away, or must be assumed away, to prove a proposition, there is a presumption of problems for the proposition itself.  We might also note that MM agree their use of the term `rational' to refer to behavior based solely on the earnings prospects is much more debatable than, say, the use of the term `rational' to refer to consistent behavior by consumers in (economic theory).  Burness, Cummin­gs, and Quirk (1980), pp.40-42.

    Stiglitz (1974) previously spoke about problems with self-fulfilling equilibria when he concluded `if individuals believe that financial policy affects firm valuation, then it will.'  Without using the term `common knowledge,' Stiglitz nevertheless implied that a common knowledge beliefs assumption was required to prevent `expectations of real returns dependent on firm financial policy.'  An interested reader would be left confused as to whether Stiglitz was assuming the very object of his proof:  debt-equity ratios are irrelevant.

    But in the final analysis, after more than two dozen separate proofs in alternative variants of the original MM model, the MM literature concluded that investor beliefs must be homogeneous (as to firm earnings and therefore risk classes) and common knowledge.  Otherwise, the MM propositions were susceptible to disproofs of the kind outlined below:

        Stiglitz (1974) seems to argue that only if investors think that financial policy will affect the `real' returns -- presumably earnings and/or dividends of a firm -- will this be a factor in determining market value.  But of course all that is required for financial policy to affect market value is the belief on the part of any investors that other investors value firms in part on the basis of the debt-equity ratio of the firm, regardless of whether the financial policy has any effect at all on earnings and/or dividends....Stiglitz is underestimat­ing the importance of the problems posed by speculative consider­ations so far as the MM proposition is concerned.  We are in what might be called an emperor's clothes situation; the MM proposition is correct only so long as all investors believe it is correct, and once any non-infinitesimal group of investors believes it is wrong, it is wrong.  In order to clear the air, and settle the matter once and for all, we would like to announce that we do not believe the MM proposition is correct.  Since we have modest (but non-infini­tesimal) investments in the market, we have now disproved the theorem.  We would be interested in alternative proofs  that bypass

        the problems posed by our approach.  Burness, Cummings, and Quirk (1980), pp. 63-65.

     We agree with the reasoning of Burness, Cummings, and Quirk, but not their conclusion.[5]  The above quotation correctly points out that in a multiperiod economy the absence of common knowledge can lead investors to speculate in a manner differing from the arbitrage predicted by the MM literature.[6]  However, even if some non-infini­tes­imal group believes debt-equity ratios matter, the MM strategy continues to lock in a sure-profit, provided the investor can hold his position long enough for earnings to be reported.  This group's actions in no way eliminate the arbitrage opportunities available in the market.  Thus, markets would not clear.  Fluctuations in security prices during the intermediate periods -before earnings are realized - represent an opportunity cost, not a real loss.[7]

5.3.  Capital Gains in the Original MM Model

Ross (1989) describes how more modern developments in arbitrage pricing have improved the original MM formula­tion of their paradigm. However, the original MM formulation is easier to understand than much of the modern theory.[8]  This section the capital gains motive in the original MM framework.  Let X_is denote the earnings of firm i in state of the world s, e_i the price of firm i's shares, and S_i the total number of shares.  Suppose firm 1 and firm 2 belong to the same risk class:  X_1s = c X_2s, across every state s.  For simplicity, set the constant c = 1, so that

                                             X_1s = X_2s / X_s.                                      (5.1)

    Let the first firm be capitalized with only equity so that its market value V₁ = E₁ = e₁ S₁, while the second firm has both debt and equity so that its market value V₂ = e₂ S₂ + B₂.  Suppose V₁ < V₂.  MM showed that owners of firm 2 shares could sell off their holdings, purchase shares in firm 1, and achieve higher cash flows for the same dollar investment.

    To see this, let some investor hold α% of firm 2's outstanding stock.  Then the income, Y_2s, the investor receives from his firm 2 shares in state of the world s is

                       Y_2s = α (X_2s - r₂ B₂)

                           = α (X_s - r₂ B₂),                                                       (5.2)

where r₂ is the market rate of return firm 2 must pay on its bonds.  Suppose the investor sells his shares in firm 2 and borrows α B₂ dollars using these combined proceeds to invest in firm 1 shares.  Let β denote the percentage of firm 1 shares he purchases.  Thus

                                   β E₁ = α (e₂ S₂ + B₂) = α V₂.

The income he receives from holding these shares in firm 1 and paying interest on his borrowed funds, again noting that X_1s / X_s, is

                              Y_1s    = β X_s - α r₂ B₂

                                      = (α V₂/E₁) X_s - α r₂ B₂

                                      = α [(V₂/V₁) X_s - r₂ B₂]                                 (5.3)

    Since V₂ > V₁ by hypothesis, then from comparing (5.2) and (5.3), Y_1s > Y_2s for every state of the world s.  Thus all investors find it profit­able, whatever their preferences toward risk, to sell their `overvalued' shares in firm 2 and acquire shares in the `undervalued' firm 1.  So far so good from an income perspective.

    A few comments are in order on the nature of the MM riskless profit.  First, as already mentioned, it assumes the investor will not have to liquidate his holdings before realizing the earnings X_s in some future period s.  Second, equation (5.3) contains the term r₂, which implies the `homemade leverage' idea that individuals can borrow at the same rate as firms.  For the arbitrage trading group of a commercial or investment bank, this assump­tion is trivial.  For individual investors in the real world, this assumption generally will not hold.

    But what happens if the investor wants to continue holding the overvalued firm 2 shares, or even buy additional firm 2 shares, because he believes he can realize an intermediate capital gain before the state is revealed in some future period.  Let E_T(V_i) denote the individual's conditional expectation for the market value of firm i in period T given the value of V_i in period 1.  If the same investor believes

                        α [E_T(V₂) - V₂] - β [E_T(V₁) - V₁] > Y_1s - Y_2s                (5.4)

then he may hold the overvalued shares with the speculative intent of realizing an intermediate capital gain.  This combined treatment of capital gains and wealth apart from income is largely missing in both the MM literature and the arbitrage pricing literature to date.

    A few comments are also necessary about condition (5.4).  The expectation operator in (5.4) would typically be conditioned on the individual's information set.  However, to assure that everyone has access to the same information, we have restricted the information set to the publicly known market values in period 1.  Even with the inequality expressed in (5.4), the fact remains that Y_1s > Y_2s, and others in the market will want to exploit this arbitrage condition.  Therefore, expression (5.4) suggests why someone may seek intermediate capital gains, but it fails to eliminate the arbitrage opportunities, a condition necessary for market clearing in the MM world.

5.4.  A Numerical Example

This numerical example will first derive conditions under which an investor may want to continue holding `overvalued' shares in a firm.  Second, we will show that under these conditions, an arbitrage opportunity exists - even if the investor still wants to hold the overvalued shares.  Consider two firms, Firm A and Firm B, with identical earnings per share of either 1, 2, 3, 4, or 5 and a sequence of events as follows.  Initially traders have a prior round equilibrium based solely on their prior beliefs for earnings.  Then after markets have cleared, Firm A changes its debt-equity ratio.  The markets reopen and trading resumes.  After markets have again closed, Firm B will announce a change in its debt-equity ratio.  Trading resumes followed by information on the actual earnings of the two firms. 

    Just as in the original MM model, let all market participants have access to the same information and share `homogeneous beliefs' in the form of a common prior over the five possible states for earnings/share.[9]  Define this common prior as Pr(1) = 0.6, Pr(2) = 0.1, Pr(3) = 0.2, Pr(4) = 0, and Pr(5) = 0.1, where Pr(s) denotes the common probability assigned to earnings/share s.

    With the prior belief distribution (.6, .1, .2, 0, .1), the market fundamental[10] of this firm's equity is 6 in period 0.  To simplify the calculations, suppose both firms' shares trade at a price/earnings ratio of 10.  Thus in the prior trading round, both Firm A and B shares clear the market at a price of $19.  Next suppose Firm A changes its debt-equity ratio, and this change is publicly announced (so everyone has access to the same information).  Suppose some non-infinitesimal group decides that debt-equity ratios matter to them, and they are willing to sell Firm A shares now for $17/share.  The rest of the market would willingly purchase from the group at that price, i.e., the group could easily find one or more trading partners.  Therefore, this group may sell their block of shares in Firm A at $17/share.  The price of Firm B shares continues to rest at $19/share.

    For the rest of the market Firm A shares now look undervalued relative to Firm B shares, and the MM arbitrage strategy suggests selling off Firm B shares and borrowing to buy Firm A shares until the two shares are again selling for the same price.  Any price discrepancy between these two firm's share prices is an arbitrage opportunity.  MM define their equilibrium as the absence of arbitrage for any investor.  This arbitrage exists by virtue of the firms having the same earnings across possible states of the world; the lack of common knowledge about investor prior beliefs in no way limits the arbitrage.  Thus, while some block of shares may trade at $17/share, this price cannot remain the market clearing price in this period.  Firm A shares will most likely be bid up again to $19/share.

    If we pursue the numerical example further, we may illuminate some features of condition (5.4).  We see from the prior distribution (.6, .1, .2, 0, .1) that investors assign the most probability to earnings/share equal to 1.  However, suppose some individual believes that other people believe the earnings/share will most likely turn out to be 3.  In that case, both Firm A and Firm B shares will eventually sell for $30/share.  Applying these numbers to condition (5.4), and noting the $2 price discrepancy, would yield

                                       .                               (5.5)

For condition (5.5) to be met,  would have to be much greater than  :  Firm A must have many fewer shares outstanding than Firm B.  Even then the condition does not guarantee any individual will actually speculate on capital gains.  Finally, if both Firms A and B did not trade at $30/share when they both report earnings of $3/share, then they would not belong to the same risk class (assuming c = 1 in equation (5.1)).  Thus another of the MM assumptions would have been violated.

5.5.  The Capital Asset Pricing Model

Like the MM paradigm, common knowledge prior beliefs are not required for the standard formulation of the Capital Asset Pricing Model (CAPM).  The CAPM, developed by Sharpe (1964), Mossin (1966), Lintner (1969), and others, equates market demand (given expected returns) with the supply of assets to solve for equilibrium prices.  In practice, finance professionals have frequently taken prices in real financial markets as given and then used the  of the CAPM to compute the return expected on individual assets (compared to the overall market's return).

    The empirical assignment of CAPM betas to firms has proven useful for guiding investment choices and evaluating the performance of firms.  In other respects, the CAPM theory had widely disconfirmed implications, e.g., that everyone should hold exactly the same proportionate mix of stocks in the risky portion of his portfolio.  Other implications of the CAPM, such as quadratic utility functions so that individuals are guided soley by the mean and variance of an asset's return or independent and normally distributed asset returns, have also been disproved.

    This section explains why investors in the CAPM paradigm do not face intrinsic uncertainty.  The CAPM assumes investor beliefs are homogeneous but not common knowledge.  Significantly, our finding thus negates the claim that theorists needed or meant to `common knowledge beliefs' for `homogeneous beliefs' all along.

    The `homogeneous beliefs' assumption has crucially simplified mean-variance asset pricing theory and related empirical works.  By homogeneous beliefs financial theorists assume

        every investor attaches the same values for the mean  and standard deviation  for the distribution of returns of every security and for the correlations  among the returns.  It follows from this that the opportunity sets of all individuals have the same proportionate shape, differing from one another by only the scale factor of individual endowed capital .  (Hirshleifer (1970), p.289)

    Although Sharpe (1964) acknowledged the homogeneous beliefs assumption was `highly restrictive and undoubtedly unrealistic,' he nevertheless defended its use.  `Since these assumptions imply equilibrium conditions which form a major part of classical financial doctrine, it is far from clear that this formulation should be rejected - especially in view of the dearth of alternative models leading to similar results.'  (Sharpe (1964), p.434)

    The two-period CAPM assumes investors know the return on assets, and this return includes both dividends and capital gains potential.  Now purchases and sales of assets by other investors clearly affect everyone's opportunity set.  Therefore, implicitly, investors in the CAPM must know the purchasing behavior of the rest of the market.  Otherwise, they could not calculate with certainty any asset's expected capital gains (losses).  Common knowledge beliefs assumptions would eliminate speculative changes in market prices due to changing beliefs.  The two-period CAPM has substituted perfect knowledge (and indeed common knowledge) about capital gains in place of an agreed common knowledge beliefs assumption over the possible states of the world.  Actually, the two assumptions are logically equivalent.

    The multiperiod CAPM, derived by Merton (1973) and extended by Breeden (1979), assumes price trajectories into the future are known.  It can be shown that intermediate period prices are common knowledge in period 0 only when individuals have agreed common knowledge priors (ACKP) about the state space.  Without ACKP individuals might not reach consensus agreement about what future prices will prevail.  In contrast, the two-period CAPM eliminates intrinsic uncertainty by assuming individuals know the correct capital gains value of assets.

    In his classic beauty contest analogy, Kenyes (1936) reveals why estimates about other investors beliefs are crucial to stock market speculation.  Since stock market investors realize other investors revise their portfolios according to changes or perceived changes in market expectations, investors need to choose stocks which they think average opinion thinks average opinion thinks will perform well.  This thought process is known as `metathinking.'  In general, investors second and third-order beliefs could differ, and consequently one would expect investors to hold differing quantities of a given asset depending on which higher-order beliefs guide their portfolio revision.

    Both the two-period and multiperiod CAPM cleverly avoid financial metathinking behavior by simply assuming investors know the assets' returns with certainty.  Once they know these returns, which may depend on higher-order beliefs or a lack thereof, they have all the information they need to minimize the variance of their portfolios for a given expected return.

    Chapter 4 showed that when investors agree without knowing they agree, intrinsic uncertainty is still present.  If CAPM investors agreed on assets' returns without knowing they agree, one might expect speculative trades to arise in the CAPM as well.  Keynes would argue that stock market investors cannot know assets' returns unless they know the higher-order beliefs of other market traders.  The CAPM simply assumes that do know the returns.  Let us turn now to a derivation of the CAPM and see where investor belief assumptions enter the model.

    The standard CAPM equation derives from the minimization problem,

                                  (5.6)

where  represents the proportion of individual k's portfolio held in asset i;  and  are the returns on assets i and j, respectively, where the return denotes the summed capital gain, dividends, and interest payments divided by the asset's price when the period commenced;   is the Lagrange multiplier associated with individual k's wealth,  is the riskless rate of interest, and  is a constant denoting individual k's expected return on his portfolio.

    The first-order interior condition stated for individual k is

                      .              (5.7)

Let  denote the fraction of total market wealth individual k holds.  Then

                                                                         (5.8)

and

                                                                                    (5.9)

where   denotes asset j's  fraction in  the  total market portfolio, and

  is the market's return on all assets taken together.  Multiplying each term in (5.7) by , summing over k, and substituting (5.8) into the first term after switching the order of summation yields

               23      (5.10)

    Next, we substitute (5.9) into (5.10) and let the variable Z denote .  Then the first-order condition becomes

                                                          (5.11)

Let  and observe that (5.11) is now

                                                              (5.12)

and solving for Z yields

                                                              (5.13)

Again substituting this value of Z into (5.11) yields the familiar CAPM equation:

                                                 (5.14)

                                                                              (5.15)

    This result only requires investors to use, e.g., the same estimate for asset i's return, .  In equilibrium, when all agents revise their portfolios according to their beliefs about the returns, they will have mean-variance efficient holdings.

    The expected value operator in equation (5.6) would likely be conditioned on each individual's information set.  However, since the CAPM assumes investors know the returns, these expected (future) values are known constants, rather than some expected value over a genuine probability distribution.  Again, the CAPM effectively eliminates any source for speculating over the beliefs of others.

5.6.  Arbitrage Pricing Theory

    The Arbitrage Pricing Theory (APT), introduced by Ross (1976) and (1977b), has now received considerable attention as an alternative to the CAPM.  Like the MM model, the APT assumes investors have common prior beliefs.  However, unlike the MM paradigm, the APT actually needs the more restrictive agreed common knowledge priors (ACKP) assumption for its validity.

    The APT in its present form yields nonunique solutions to the market pricing model.  Without common knowledge assumptions, the APT yields individual asset pricing equations that cannot guarantee that the rest of the market uses the same or even similar pricing relationships.

    The logic behind the APT might be expressed as follows.  The returns on a subset of assets in the market are governed by an M-factor model.  Unfortunately, neither the list of factors, nor the specification, nor the subset of assets are ever definitively identified in the APT literature.  This lack of a logical structure has led to numerous empirical problems, but we will focus here on the theoretical model.

    The APT assumes investors have `homogeneous' beliefs about the factor model generating asset returns.  As Roll and Ross (1980) have suggested, `[t]he key point in aggregation is to make strong enough assumptions on the homogeneity of individual anticipations to produce a testable theory.  To do so with the APT we need to assume that individuals agree on both the factor coefficients, , and the expected returns,  '  Ross (1976) at page 356 indicated that the topic of investor belief assumptions is `one of the most difficult and important areas for future research.'  Ross concluded that relaxing the homogeneous expectations assumption would require study of disequilibrium dynamics and movement from ex ante expectations to ex post observations.

    Since its introduction in 1976, the APT has been refined and extended by Huberman (1982), Chamberlain and Rothschild (1983), and Ingersoll (1984), among others.[11]  Subsequently, Dybvig (1983) and Grinblatt and Titman (1983) independently derived upper bounds on the deviations from APT-predicted prices.

    All of these papers begin with the same underlying assumption that asset returns are generated by a factor model.[12]  Consider a market with Z risky assets and a riskless asset with return .  There exists a subset of the assets numbered 1 through T whose returns are governed by the factor model

                                            (5.16)

where  = the return of asset j;  = the information set, upon which the expected value is conditioned;  = sensitivity of asset j's return to movements in factor k;  = mean zero idiosyncratic risk component of the j^th asset, j = 1, ..., T.  Assume that the error terms, , are independent of the factors, the returns, and of each other.  Therefore,  = 0 for all .  Further assume that T > M, in order to ensure that the system of equations is determined.

    Our model depicts the standard form of the APT with the addition of the information set .[13]  The theoretical intuition behind the APT is that for some subset of the assets under consideration, the returns will be a linear function of the factors in equation (5.16).  In practice, these factors are `grand aggregates' such as the employment level or the interest rate.  Conversely stated, the APT implies that for a large enough subset of assets, the only possible common determinants of their returns could be the various grand aggregates, although empirical studies reveal little agreement on what these factors should be.

    Each individual formulates his own measure of the market's expected return from assets.  Under the homogeneous beliefs assumption investors in the APT, investors might agree on the returns and the M-factor model without knowing they agree.  The intrinsic uncertainty in the APT can lead investors to question what factors, functional relationship, or information the rest of the market employs in deciding market values for assets.

    An investor in the APT knows what factors and relationships he personally believes will affect asset returns, but he faces risk over the factors and relationships others feel are important.  Consider now the following three factor equations.  All three are possible candidates for the unknown factor model employed by the market, and all three are perfectly consistent with the existing assumptions of the APT.

    First, we have

                                         (5.17)

where the number of factors M-K is less than the M factors in (5.16).  Second,

                                              (5.18)

where  and third