Homogeneous Beliefs and Speculation
in Asset Pricing Paradigms

MICHAEL A. S. GUTH, Ph.D., J.D. 
Homogeneous Beliefs and Speculation
in Asset Pricing Paradigms
[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
marketlevel 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 pricetoearnings ratio for each
firm's stock. This chapter examines
whether ModiglianiMiller (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 ModiglianiMiller (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 debtequity 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 twoperiod (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 wellknown aggregation problems with diverse individual beliefs
have somehow sorted themselves out, and the economy starts in a marketclearing
equilibrium.
Our
analysis can be distinguished from the older, precommon 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 debtequity 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 debtequity 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 MMstyle
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
MMstyle 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 debtequity 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 selffulfilling 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 debtequity 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.4042.
Stiglitz
(1974) previously spoke about problems with selffulfilling 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: debtequity 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 debtequity 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
noninfinitesimal 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 noninfinitesimal) 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. 6365.
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 noninfinitesimal
group believes debtequity ratios matter, the MM strategy continues to lock in
a sureprofit, 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 V_{1}
= E_{1} = e_{1} S_{1}, while the second firm has both
debt and equity so that its market value V_{2} = e_{2} S_{2}
+ B_{2}. Suppose V_{1}
< V_{2}. 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}
 r_{2} B_{2})
= α (X_{s}  r_{2} B_{2}), (5.2)
where r_{2} 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 B_{2}
dollars using these combined proceeds to invest in firm 1 shares. Let β denote the percentage of firm 1 shares he
purchases. Thus
β E_{1}
= α (e_{2}
S_{2} + B_{2}) = α V_{2}.
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 r_{2} B_{2}
=
(α V_{2}/E_{1})
X_{s}  α r_{2} B_{2}
=
α [(V_{2}/V_{1})
X_{s}  r_{2} B_{2}] (5.3)
Since V_{2}
> V_{1} 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 r_{2}, 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}(V_{2})
 V_{2}]  β [E_{T}(V_{1})  V_{1}] > 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 debtequity ratio.
The markets reopen and trading resumes.
After markets have again closed, Firm B will announce a change in its
debtequity 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 debtequity ratio, and this change is publicly announced (so
everyone has access to the same information).
Suppose some noninfinitesimal group decides that debtequity 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 meanvariance 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
twoperiod 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
twoperiod 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 twoperiod
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 thirdorder beliefs could differ, and consequently one would expect
investors to hold differing quantities of a given asset depending on which
higherorder beliefs guide their portfolio revision.
Both the
twoperiod 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 higherorder 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 higherorder 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
firstorder 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
firstorder 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 meanvariance 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 Mfactor 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
APTpredicted 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 Mfactor
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 MK is less
than the M factors in (5.16).
Second,
_{} (5.18)
where _{} and third
_{} (5.19)
The intrinsic uncertainty present in the original
formulation of the APT would not allow any investor to exclude the possibility
that other investors in the market employ any or all of these three factor
models.
The
first example (5.17) altered the number of factors believed to determine asset
returns. Shanken (1982) has pointed out
that financial economists have yet to determine empirically the precise number
of factors in the APT. The statistical
data on asset returns often reveal the great majority of variance to be
explained by a single factor, and that the other factors though containing
significant coefficients nevertheless account for a negligible amount of the
variance.[14] If both the APT and the CAPM explain the
variance in asset returns with primarily one factor, then these empirical
observations raise questions about the utility of switching from the CAPM to
the APT.
The
specification problems encountered in trying to test the APT support the notion
that the APT is not common knowledge, even among financial econometricians or
arbitrage pricing specialists. The
econometricians may have homogeneous beliefs; they may agree that an Mfactor
linear model governs asset returns. Yet
uncertainty over what variables and factors the rest of the market might regard
as relevant has led each of them to test for the significance of factors
presumably not contained in their original Mfactor model. The uncertainty over relevant factors faced
by empirical researchers mirrors the uncertainty faced by investors in the APT
model even under the homogeneous beliefs assumption.[15]
The
second example (5.18) has replaced the factor coefficients, because although
investors may objectively agree on the values of these coefficients, they do
not know the values other investors assign to these coefficients. Again the empirical literature has arrived
at no consensus on the factor coefficients.
This lack of empirical agreement might well be explained by theoretical
intrinsic uncertainty that investors do not know, and know everyone else knows,
what the true coefficients are in the market model.
Finally,
the third factor model (5.19) illustrates this same point for the expected
returns. When _{} in the third model,
an investor believes he is better informed than at least some subset of the
market; conversely, if _{}, an investor feels the market has additional information
upon which to condition their expectation for the asset's return.
This
section has been based on work published by Guth and Philippatos (1987). The latter work contains propositions that
state (1) the sum of squared deviations from (5.16) is not bounded above as the
number of assets increases; (2) the APT has multiple equilibria consistent with
the homogeneous beliefs assumption; and (3) that imposing ACKP will yield a
unique expression for the APT.
Finally,
we conclude the section by noting that the standard APT return expression is a
linear equation, and the APT cannot account for nonlinear relationships through
interpreting the factors in the APT as nonlinear functions of real variables. Without common knowledge assumptions, any
individual could personally believe these factors affect returns in a linear
form but feel the rest of the market employs a nonlinear relationship.[16]
5.7.
Heterogeneous Beliefs in the MM and APT Paradigms
Stiglitz (1969, 1974) and Stambaugh (1983)
maintained that the MM and APT models, respectively, are valid even if
investors have heterogeneous beliefs.
We will consider their arguments sequentially.
In
claiming to have generalized the original MM framework to include heterogeneous
beliefs, Stiglitz asserts
No
assumption about the source of uncertainty is required. In particular, the distinction between
technological uncertainty and price uncertainty...is of no consequence here.
...The argument of the proof does not require that individuals have the same
expectations. The only agreement in
expectations that is required is that the firm will not go bankrupt in any
state of nature. Stiglitz (1974),
p.861, 861n.
Stiglitz's claim that price uncertainty has no
more consequence than technological uncertainty sounds suspicious. He seems to have overlooked speculative
investments based on the unknown beliefs of others. But Stiglitz (1988) subsequently qualified his own remarks.
It
is often the unstated rather than the stated assumptions of a model which are
critical, and so it is in the case of the MM theorems. A critical unstated assumption is that all
market participants have full and equal information concerning the returns to
the firm. However, the existence of
asymmetric information gives rise to two problems: current owners may wish to convince potential borrowers that the
firm is worth more; and managers can take actions which affect the returns to
those who provide capital. Stiglitz
(1988) pp. 123124.[17]
Given
the caveats that investors (1) assign zero probability to any firm going
bankrupt and (2) have full and equal information concerning the returns to the
firm, we will impose heterogeneous beliefs on Stiglitz's model and see whether
the MM irrelevancy theorems still hold.
Table 5.1 defines a subset of the variables used by Stiglitz.
Table 5.1. List of Relevant Variables in Stiglitz's
(1974) Model
_____________________________________________________________
Symbol Definition
_____________________________________________________________
P^{ j}(θ(t)) the probability investor j assigns
to state θ(t).
_{} the profits to
firm i at time t in state of the world _{}.
p(t,τ) the
price at time t in state θ(t) of a bond which promises to pay 1 dollar at time τ.
B_{i}(t,τ) the number of bonds for firm i
outstanding at the end of period t with maturity at time τ.
B^{ j}(t,τ) the j^{ th}
individual's ownership of bonds maturing at time τ.
E_{i}(t) the
value of the shares outstanding in the i^{ th} firm at the end
of period t.
α_{i}^{j}(t) = E_{i}^{j}(t)/E_{i}(t),
is the fraction of the i^{ th} firm held by the j^{ th}
individual at the end of period t.
____________________________________________________________
Two
market clearing conditions characterize an equilibrium. The first states that ownership of shares in
the i^{ th} firm equals the value of its equity:
_{} (5.20) 
The second equates demand and supply of bonds at
each maturity, _{}:
_{} (5.21) 
Note that condition (5.20) is equivalent to _{}
Stiglitz's
main theorem then states if (1) no firm goes bankrupt in any state of nature,[18]
(2) the economy contains a liquid market for perfectly safe bonds of all
maturities, (3) all firms have chosen their financing policy, (4) given a set
of feasible consumption paths, every consumer always selects the same
consumption path, and (5) the market is in a general equilibrium with all
markets clearing; then there exists another general equilibrium in which any
firm or group of firms has changed its financial policy, but in which the value
of the firms and the price of all bonds of all maturities are unchanged. Individuals have adjusted their portfolios
as follows:
_{} (5.22) 
i.e, each investor alters his holdings of bonds
by exactly his stockholder's share of the change in debt of each maturity of
all firms, and
_{} (5.23) 
or equivalently, each investor changes his equity
holdings in proportion to the firm's change in total equity capital. Let us consider the two most natural forms
of heterogeneous beliefs.
Case
1. For two or more representative individuals i and j, P^{
i}(θ(t)) _{} P^{ j}(θ(t)). The two individuals would therefore also
disagree on the expected profits to each firm, since these profits are
statecontingent. However, the
individuals then disagree as to the returns to firms. Clearly, Stiglitz's model cannot handle the problems created by
this situation, for it implies that the two individuals disagree on the fair
price for virtually all, if not all, of the firms in the economy. If these two individuals represent the
beliefs of sizable fractions of the population, then markets will never clear.
Alternatively,
we could say the disagreement over profits to firms across states of the world
violates Stiglitz's (1988) assumption of `full and equal access to information
on firm earnings.' Under this
interpretation, Stiglitz's model again fails to accommodate the heterogeneous
beliefs P^{ i}(θ(t)) _{} P^{ j}(θ(t)) by
construction, rather than by implication.
Case 2. Assume
the two individuals i and j agree on the probability distribution
over states of the world so that P^{ i}(θ(t)) = P^{ j}(θ(t)) for all _{}. However, i
and j have different priors as to the profits of every firm n for
each state of world: _{}, where _{} represents the
expectation operator for each individual.
Normally the expected value of a variable would be conditioned on each
individual's information set. However,
to adhere rigidly to Stiglitz's `full and equal access' requirement, we will
assume no one has received any information about earnings. The individuals are merely endowed with
heterogeneous priors as to how profits are distributed across states of the
world. Markets will never clear in this
economy, because the individuals cannot agree on fair prices for stocks. One or more of the individuals will find
mispriced stocks and want to buy more claims to those stocks than are available
to the market; others may want to sell or supply more shares at that price than are available. Thus condition (5.20) will not be met.
In the
Stiglitz (1974) multiperiod MM model, if investors maintain heterogeneous
beliefs about the probability distribution over states of the world [P^{
i}(θ(t)) _{} P^{ j}(θ(t))] or the
distribution of profits to firms across states of the world {_{}}, then markets will never clear. Thus Stiglitz's restated irrelevancy theorem has no consequence
even for his own economy.
Stiglitz
also abandoned the MM `risk class' concept in favor of assumptions that
consumers always select the same consumption path and expectations independent
of financial policy. Stiglitz requires
these two assumptions to motivate investors to undo the leverage or other
financial policy change adopted by firms.
By assuming that individuals believe financial policies have no effect
on firm valuation, Stiglitz practically assumes the very object of his proof.
In
contrast, we avoided this unwonted assumption by using the risk class
concept. Section 5.3 shows the MM
arbitrage locks in a sure profit even if some `noninfinitesimal group,' or the
rest of the market for that matter, believes financial policies matter. Without the risk class concept, Stiglitz is
left with a theorem and a framework that applies only if the economy somehow,
magically, reaches equilibrium.
Homogeneous beliefs about each firm's earnings across states of the
world clearly define arbitrage opportunities.
Without homogeneous beliefs, individual investors could see arbitrage
opportunities directly in conflict with one another.
Turning
now to the APT, Stambaugh (1983) has argued that the APT applies even when
investors have heterogeneous beliefs.
The preceeding arguments concerning heterogeneous beliefs in Stiglitz's
reformulation of the MM paradigm apply to Stambaugh's work as well. In particular, if two or more people
disagree over the number of factors, or the returns to firms, or other
specifications in the APT, somone may see arbitrage opportunities in prevailing
market prices. In that case, markets
will not clear.
Using a
particular factor model (5.16), an individual may decide a given stock is
overvalued. Another individual decides
the same stock is undervalued using a different factor model (5.17). At any given price, one or both of the
individuals will continue to show positive or negative excess demand.
How do
we determine who is correct? Eventually
when the state of the world is revealed and actual earnings become common
knowledge, we will know which one correctly valued the firm. Perhaps neither did. After earnings are announced and their effect
on the firm's stock price is revealed, the two would no longer have
heterogeneous beliefs over that firm's earnings. They may continue to employ different factor models and continue
to disagree on future earnings.
Since
earnings and the state of nature could prove either of these individuals are
correct, each faces some risk  whether he chooses to recognize it  that his
own valuation may be wrong. In the APT
with heterogeneous beliefs, everyone can personally believe he sees a riskless
arbitrage opportunity, when in reality his investment amounts to risky
speculation. Only an individual who
assigns zero probability to his Mfactor model being incorrect would be
willing to serve as a price dictator and force prices to match his own
valuation model.
In game
theory terminology, the heterogeneous belief investors can each act as
Stackelberg leaders. Without other
parties agreeing that their own factor models are inferior and serving as
followers, the disparate investors would continually bid against each
other. Prices would continually move up
and down with the bids and counterbids.
The resulting disequilibrium dynamics is known as Stackelberg warfare.
5.8.
Conclusions and Issues for Future Research
This chapter has examined three asset pricing
paradigms and found that common knowledge beliefs are not required for the MM
model, implicitly assumed in the CAPM, and definitely required for the validity
of the APT. The MM world depicts a
stylized view of financial markets with certain simplifying assumptions, for
example, no bankruptcy, firm risk classes, and personal borrowing at the same
rate as corporate borrowing. In an
effort to eliminate intrinsic uncertainty from their model, MM inserted an
additional `imputed rationality' assumption.
As indicated in Section 5.2, the only previous paper to take notice of
this missing assumption concluded that the MM results were invalid (unless
investors had agreed common knowledge beliefs). We can relax the `imputed rationality' assumption, which is tantamount
to assuming investor beliefs are not only homogeneous but common knowledge, and
the key results of the MM model are left intact. With this analysis, financial economics appears to have taken a
step towards relaxing the restrictions on investor beliefs that has plagued the
development of most asset pricing theories.
The
absence of common knowledge creates an asymmetric information environment, but
the MM arbitrage still holds under this particular dimension of
uncertainty. Any individual can exploit
cash flow arbitrage given sufficient resources and a suitably long investment
horizon. Even if some
`noninfinitesimal' segment of the market believes debtequity ratios matter,
the MM strategy continues to lockin a sure profit. This market segment can be used as a money pump by the rest of
the market.
The CAPM
assumes investors know the future returns on assets. This assumption effectively eliminates
capital gains uncertainty and the source of speculation over the unknown
beliefs of others. This strong
assumption implies that investor beliefs are common knowledge. The APT employs homogeneous belief
assumptions in an attempt to aggregate individual factor models into a market
expression. However, this attempt falls
short due to intrinsic uncertainty inherent in the APT.
Since
the APT uses MMstyle arbitrage, why would investor beliefs have to be common
knowledge in the APT but not in the MM world itself? The answer may emerge from the response in Endnote 5 to the
disproof the capital structure irrelevancy proposition. MM arbitrage focuses on identical cash flow
streams; the APT emphasizes a murky Mfactor model. Thus the objects of uncertainty are
diverse. It would be like trying to
compare the volume of ketchup in two bottles (an objective, measurable fact)
with the ingredients that make ketchup taste good (an inherently subjective
concept). Not all arbitrage theories
are created equal, even though they employ some variant of MM arbitrage.
In
contrast to Stiglitz (1988), we find that the `risk class' concept is important
for MMstyle arbitrage, even in Stiglitz's general equilibrium framework. Without risk classes and homogeneous
beliefs, we have no mechanism to eliminate arbitrage opportunities and clear
markets. Without market clearing the
applicability of Stiglitz's theorem, on moving from one general equilibrium to
another, vanishes. By reinserting the
risk class concept in Sections 5.3 and 5.4, we do not need to restrict
individuals to adopt the same consumption path in every equilibrium or to form
expectations independent of financial policy.
Assuming
investors have `full and equal access to information on earnings' effectively
eradicates, by construction or by implication, the two most natural forms of
heterogeneous beliefs: differing
probabilities for states of the world or differing earnings expectations for a
firm in a given state of the world.
Thus, while not stated in his theorem, Stiglitz effectively assumes
homogeneous investor beliefs like the original MM paradigm.
At least
one issue remains unresolved. If an
individual sees an arbitrage opportunity in either the MM or APT and executes
the income arbitrage, what impact does he have on market prices? Under competitive assumptions, individuals
take prices as given and have negligible impact on market prices. However, various arbitrage pricing models
employ the `price dictator' concept, in which any individual can sell an
infinite quantity of some asset, if needed, to force the asset's price to
adjust and thereby eliminate the arbitrage profit. If we assume no firm goes bankrupt, should we assume no
individual can go bankrupt as well? If
so, then an individual could borrow or sell infinite quantities of assets.
Intrinsic
uncertainty is not merely a technical point but actually an important feature
of real world financial markets. It is
all too easy to fall into the trap of labeling certain assets as `undervalued'
or `overvalued' or even `a free lunch,' without appreciating the underlying
risks involved. As a case in point,
Japanese warrants in 1994 appear to be undervalued. They are undervalued in comparison to German warrants. They are undervalued in comparison to
shorter term Japanese call options, and they are undervalued in comparison to
their historical implied volatilities.
Nevertheless,
only a foolhardy person would rush to buy `undervalued' Japanese warrants based
on these facts. The reasons were
outlined in Sections 5.2 and 5.3. The
market contains intrinsic uncertainty, which forces every rational investor to consider
the resell potential of his assets based on changes in future market
valuations. The Japanese warrant market
could remain in a longrun `undervalued' slump for years. Any financial institution that tries to
execute income arbitrage could have its resources tied up with no return or
worse, while it waited for earnings of underlying stocks to grow. To profit from Japanese warrant trading,
arbitrageurs must accurately forecast not only the earnings of the underlying
firms, but when these increases in earnings will be reflected in the longterm
warrant prices.
There
exists no perfect MMstyle arbitrage with Japanese warrants. For example,
German warrants are correlated to Japanese warrants, but anyone using German
warrants as a hedge is exposed to risk that this correlation will change over
time. Arbitrage desks of investment
banks are in the business of managing risks, not simply pursuing `free
lunches.' The risks associated with the
absence of common knowledge can be eliminated in the MM world; however, this
perfect arbitrage strategy is rarely available in real financial markets. It is not surprising then that the capital
gains motive affects asset prices, which are governed by market sentiment and
other intrinsic factors aside from real earnings.
Having
considered the need for common knowledge beliefs assumptions in three asset
pricing paradigms, we might wonder how common knowledge assumptions affect the
canonical arbitrage pricing model. The
next chapter covers this topic. We
prove a series of `no price exists' propositions and relate the lack of common
knowledge to the amount of information about higherorder beliefs revealed by
prices. Chapter 3 emphasized that
incomplete markets force everyone to speculate in the sense that optimal consumption
bundles cannot be obtained from the available market instruments. Consequently, everyone plans to
retrade. Chapter 6 shows that
speculation based on uncertainty over other traders' (changing) beliefs will
tend to be the norm rather than the exception.
Notes
1. I would
like to thank Merton Miller, Ronald Shrieves, Robert Jarrow, and Jack
Hirshleifer for helpful comments on earlier drafts. None of the above are responsible for any errors that remain.
2. See also
Bhattacharya (1979), Myers and Majluf (1984), and Stiglitz (1988).
3. Indeed no
mechanism in any of the major asset pricing theories transmits this information
(about the beliefs of others) to individual investors.
4. Grinblatt
and Titman (1983) and Stiglitz (1974) claim to have developed equilibrium
formulations of the APT and MM paradigms, respectively. Their models still rely on arbitrage for
price adjustments and the absence of arbitrage for market clearing.
5. The
answer to the ``emperor's clothes disproof'' is as follows. If an 8 oz. bottle of ketchup sells for $1
and someone offers to buy 16 oz. of ketchup for $3, the market will supply the
demander with as much ketchup at that price as he is willing to purchase. The fact that some `noninfinitesimal' group
steps forward to say the volume of ketchup (read earnings of a firm in a given
risk class) in two 8 oz. bottles is not equivalent to the volume (read earnings
of another firm in that same risk class) in a 16 oz bottle does not preclude
the rest of the market from exploiting this arbitrage opportunity. In fairness, we should note that Burness,
Cummings, and Quirk (1980) were aware of similar arguments and found them
unpersuasive: `in effect, this reduces
things back to a twoperiod model, since market values in intervening periods
(between now and doomsday) are irrelevant.' (p.62) Perhaps the matter will remain a philosophical disagreement.
6. An
investor motivated by capital gains may buy into an `overvalued' stock at
present if he believes he can resell it at a profit. Alternatively, an investor may choose to sell `undervalued'
stock at present with the speculative intent of repurchasing the stock at an
even lower price. From a wealth
standpoint, this investment strategy is perfectly rational.
7. Durand
(1989) has noted that pursuing speculative capital gains is more risky than
collecting a known dividend. `But with
volatile stock prices, capital losses....are a real possibility. Dividends represent a bird in the hand;
capital gains  representing only one bird in the bush if MM are to be taken
literally  may never materialize.'
8. See,
e.g., the vector analysis in DeMarzo (1988).
9. The
numerical example also applies to the case of heterogeneous priors.
10. The
market's collective judgement about the value of the firm measured as an
expected value of future earnings.
11. However,
these analyses were complex and their derivations of the APT formula lacked
economic intuition, as articulated in the criticism by Shanken (1982).
12. Ross
(1976) and Huberman (1982) went to some length to specify the necessary
conditions that ensure arbitrage opportunities would apply to any asset
deviating from the formula of the APT.
These conditions include von NeumannMorgenstern utility functions for
investors, as well as assumptions on the factor and error distributions. Yet, as we show in this section, all these
restrictive conditions failed to produce true arbitrage opportunities in the
absence of common knowledge.
13. Note
that we have assumed nothing about investor preferences of the objective
function they seek to maximize.
14. Trzcinka
(1987) has pointed out that the empirical tests of the APT often find seven or
eight significant factors, but one factor predominantly accounts for the
variance with the remaining portion split among the other factors tested.
[I]t
is clear from our analysis of the eigenvalues and our chisquared tests that
the question of how many factors (apply to the APT) is far from settled. Our partial and unsatisfying answer to this
question is `at least one.' In spite of
this ambiguity, it is useful to know that the number probably is not zero. Trzcinka (1987), p.368.
In
private correspondence, Richard Roll has disagreed with my characterization of
the APT empirical literature as having found that one factor explains most of
the return variance. That conclusion
would be expected for empirical models of the return on a large diversified
portfolio. However, for individual
stocks, he has found that the first factor generally accounts for 20% of the
total variance while the second and higher factors usually account for another
15 to 20 percent.
15. A
similar problem crops up in trying to specify the factors relevant to an
individual stock's return from a model applicable to a subset of stocks.
[T]hese
derivations of the APT have been relatively complex and lacking in economic intuition. Furthermore, the pricing equation obtained
from them does not necessarily hold for any individual asset, even
approximately. This has led some
researchers to conclude that the empirical tests of Roll and Ross (1980), Chen
..., Oldfield and Rogalski..., and Brown and Weinstein... are not valid tests
of theory....Our model suggests that the existing tests of the APT are
appropriate in an economy with nontraded assets. Grinblatt and Titman (1983), pp.497, 505.
Dhrymes,
Friend, and Gultekin (1984) found that standard _{} tests of the APT
require an increasing number of factors as more and more securities are
analyzed. See also Roll and Ross's
(1984) reply.
16. Chamberlain
and Rothschild (1983) prove that a necessary and sufficient condition for
returns to be generated by a Kfactor linear model is that K
eigenvalues of the covariance matrix of returns grow large as the number of
securities increases.
17. Stiglitz's
quotation reiterates what we stressed in Section 5.2: asymmetric information in the MM literature immediately
translates into insider (manager or current shareholder) incentives. In fact, it applies to anyone who is
concerned with wealth in addition to income.
18. Smith
(1972) and Hellwig (1981) showed how the MM propositions could be extended in
an economy that included possible firm bankruptcy. We omitted a discussion of their models in this chapter, because
it entails additional specifications for margin contracts.
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Comment
In reading through (an earlier draft of) the
Chapter (5) you gave me, I was reminded of some discussions that Franco
Modigliani and I had many years ago on a possible problem with our proof very
similar in spirit to the one you point out.
We did not use the term `common knowledge,' of course; that term had not
been invented yet. But we were very
definitely aware of the problem. The
missing assumption was noted and introduced not in the 1958 leverage paper, but
in the 1961 dividend paper (see p. 427 and especially footnote 24). The treatment there is admittedly not very
rigorous by present day standards. But
at least we sensed an incompleteness that, somewhat to our surprise, no one
seems to have followed up prior to your paper.
Merton H. Miller
Graduate School of Business
University of Chicago
The j^{
th} individual's wealth is defined as
_{} (5.22) 
Combining (5.22) with the accounting identity
_{}

yields
_{} (5.23) 
Finally, the j^{ th} individual's
consumption in period t is defined as
_{}
