Model Uncertainty, Limited Market Participation
and Asset Prices
H. Henry Cao, Tan Wang and Harold H. Zhang∗
This Version: October 16, 2002
Abstract
We demonstrate that limited market participation can arise endogenously in the
presence of model uncertainty. Our model generates novel predictions on the relation between limited market participation, the equity premium, and the diversification
discount. When the dispersion in investors’ model uncertainty is small, full market
participation prevails in equilibrium. In this case, the equity premium is unrelated
to the model uncertainty dispersion and a conglomerate trades at a price equal to
the sum of its single segment counterparts. When the model uncertainty dispersion
is large, however, some investors optimally choose to stay sidelined in equilibrium. In
this case, the participation rate and the equity premium decrease with the model uncertainty dispersion. This is in sharp contrast to the understanding in the existing
literature that limited market participation leads to higher equity premium and helps
to resolve the equity premium puzzle. Moreover, when limited market participation
occurs, a conglomerate trades at a discount relative to its single segment counterparts.
The discount increases in model uncertainty dispersion and is positively related to the
proportion of investors not partcipating in the markets. A conglomerate merger also
reduces the proportion of market participants. Our finding offers a new explanantion
for the diversification discount from the asset pricing perspective.
∗
H. Henry Cao and Harold H. Zhang are with the Kenan-Flagler Business School, University of North
Carolina, Chapel Hill, NC 27599-3490, USA, caoh@kenan-flagler.unc.edu, zhangha@kenan-flagler.unc.edu;
and Tan Wang is with the Faculty of Commerce and Business Administration, University of British Columbia,
Vancouver, British Columbia, Canada, V6T 1Z2, [email protected]. The authors are grateful to
seminar participants at the economics department of the University of North Carolina at Chapel Hill and
the finance group of the University of North Carolina at Charlotte, and University of California, Riverside
for helpful comments. Tan Wang thanks the Social Sciences and Humanities Research Council of Canada
for financial support.
1
Introduction
It has been well documented that a significant proportion of the U.S. households does not
participate in the stock markets. For example, based on the 1984 Panel Study of Income
Dynamics (PSID) data, Mankiw and Zeldes (1991) find that among a sample of 2998 families,
only 27.6% of the households own stocks. For families with liquid assets of $100,000 or
more, only 47.7% own stocks. More recent surveys show that even during the 1990s with
the tremendous growth of the U.S. stock markets, such limited market participation still
exists. For instance, the 1998 Survey of Consumer Finances shows that less than 50% of
U.S. households own stocks and/or stock mutual funds (including holdings in their retirement
accounts).
The purpose of this paper is to examine in a general equilibrium framework the implication of endogenous limited market participation for the equity premium and the diversification discount in the presence of heterogeneous model uncertainty among investors.
Specifically, we demonstrate that (1) the equity premium can be lower in the economy in
which there is limited market participation than in the economy in which there is full market
participation; and (2) when there is limited market participation, the value of firms traded
as a bundle is lower than the sum of value of the firms traded separately, i.e., there can be
a diversification (conglomerate) discount.
Several studies document that limited market participation is related to the equity premium. For instance, Mankiw and Zeldes (1991), Vissing-Jorgensen (1999), and Brav, Constantinides, and Geczy (2002) empirically demonstrate that if the subsample of stockholders’
observations is used, the standard asset pricing models such as Lucas (1978) and Breeden
(1979) can generate higher equity premium with a reasonable risk aversion coefficient or can
match the observed equity premium using a much lower risk aversion coefficient than the one
used in Mehra and Prescott (1985). Basak and Cuoco (1998) also show theoretically that
the equity premium increases as more investors are excluded from the risky asset markets.
The limited market participation in these studies, however, is exogenously determined rather
than arisen endogenously from the economy.
A related issue that remains unexplored in existing studies is how limited market partic1
ipation interacts with the market incompleteness as the number of securities reduces when
firms are combined into conglomerates through mergers and acquisitions. It would be interesting to determine how this interaction between limited participation and market incompletion affects asset prices of combined firms. A number of studies document that conglomerates
and multi-division firms tend to trade at a discount or negative excess value relative to their
single segment counterparts (see Lang and Stulz (1994), Berger and Ofek (1995), and Rajan,
Servaes, Zingales (2000)). Berger and Ofek (1995) in particular show that conglomerates
exhibit on average negative excess value of 10-15%. The flip side of the conglomerate discount is that spin-offs can increase firm value. Hite and Owers (1983), Miles and Rosenfeld
(1983) and Schipper and Smith (1983) have documented a 2-3% increase in shareholder value
around the corporate spin-offs. Rajan, Servaes, Zingales (2000) and Scharfstein and Stein
(2000) argue that conglomerate discount can be attributed to agency problems such as rent
seeking at the divisional level or empire building by top management. These theories are all
from corporate finance perspective.1
In terms of methodology, our paper is related to the growing literature on model uncertainty that follows the Knightian approach.2 In his seminal work, Knight (1921) makes a
distinction between risk and uncertainty. A random variable is risky if its probability distribution is known, and is uncertain if its distribution is unknown. For example, betting red
on a roulette is risky but there is no uncertainty because the odds are known. However, an
investor who purchases 100 shares of Microsoft stock is faced with uncertainty, in addition
to risk, as the distribution of the return on the Microsft stock is not known for sure. Ellsberg
(1961)’s experiments show that indivduals are repelled by unknown probabilities, which is
inconsistent with the expected utility model. Knightian uncertainty is operationalized by
Gilboa and Schmeidler (1989) using maxmin expected utility. In Gilboa and Schmeidler
1
Miller (1977) argues informally that the short sale constraint and differences of opinion can explain the
diversification discount. Naik, Habib and Johnsen (1999) offer an explanation of spin-off premium based on
higher incentive to collect information associated with spin-offs which reduces the risk premium.
2
The most common approach of modelling imperfect knowledge of the model and parameters is the
Bayesian method (Bawa, Brown and Klein (1979), Kandel and Stambaugh (1996), and Pástor (2000)). In
the Bayesian approach, model misspecification comes in the form of parameter uncertainty which is incorporated in investor’s decision making using a single prior distribution about the uncertain paramters. This
reduces investor’s uncertainty to a single probability distribution. However, as discussed later, Ellsberg’s
experiments suggest that some investors’ uncertainty cannot be expressed using a single probability distribution. Moreover, in the Bayesian approach, an investor’s decision is highly sensitive to his prior in portfolio
choice decisions as shown in a recent study by Wang (2002). In this case an investor may use a set of priors,
which reduces to our approach.
2
(1989), decision makers examine a consumption bundle according to the minimum expected
utility over a set of probabilities.3 In this paper, we adopt the maxmin expected utility
approach.4
We perform our analysis in a tractable single period mean-variance framework with
heterogeneous agents and model uncertainty. Investors can trade two risky assets and one
risk free asset.5 When making their optimal investment decisions, investors are uncertain
about the true model of the risky asset payoffs: they have accurate estimate of the variance
covariance matrix of the payoffs, but are uncertain about the true mean payoffs. Moreover,
investors are heterogeneous in their level of uncertainty. The crucial feature of our framework
is that for each agent there is a region of prices over which the agent will neither buy nor
sell short the assets. Investors who are more uncertain about asset mean payoffs have larger
nonparticipation regions than investors with less uncertainty on the mean payoffs. As a
result, in equilibrium the investors with high uncertainty choose not to participate in the
markets, giving rise to limited market participation.6
Several models have been proposed to explain why limited market participation may
exist (Allen and Gale (1994), Williamson (1994), Haliassos and Bertaut (1995), VissingJorgensen (1999), and Yaron and Zhang (2000)). Those models focus on how entry costs
and/or liquidity needs create limited market participation. Our study offers an alternative
explanation for limited market participation based on investors’ heterogeneity in model uncertainty. Our model further helps to explain why families with substantial liquid assets (over
$100,000) may not participate in stock markets. The theoretical prediction of our model on
3
See also Dow and Werlang (1992), Epstein and Wang (1994, 1995), Anderson, Hansen and Sargent
(1999), Maenhout (2000), Uppal and Wang (2001), Chen and Epstein (2001), Epstein and Miao (2001),
Routledge and Zin (2001), Kogan and Wang (2002), Liu, Pan and Wang (2002), among others. Our paper is
closely related to Kogan and Wang (2002) in its modelling framework but we consider an equilibrium model
with heterogeneous agents.
4
For alternative specifications of Knightian uncertainty, see Schmeidler (1989) and Bewley (1986). In
Schmeidler (1989), an agent evaluate a choice according to its expected utility using a capacity (Choquet
expected utility with non-additive probability). Bewley (1986) considers incomplete preferences where a
consumption bundle dominates another if and only if its expected utility dominates the other using all
probabilities in a set.
5
The model can be easily extended to include multiple risky assets.
6
Our model is related to models with heterogeneous beliefs such as Williams (1977). However, for a
model like Williams (1977) to generate limited participation, one has to assume that investors who are
sidelined believe that the equity premium on risky assets is zero. In addition, models with heterogeneous
beliefs cannot generate the relation between limited market participation, equity premium and diversification
discount uncovered in our study.
3
market participation is consistent with the empirical evidence documented in Blume and
Zeldes (1994) and Haliassos and Bertaut (1995) that investors who are less averse to the
randomness of financial assets are more likely to participate in the stock markets.
The intuition for the implications of limited market participation on the equity premium
can be explained as follows. In our model, equity premium has two components, the risk
premium and the (model) uncertainty premium. When the dispersion of model uncertainty
among investors is high, only investors with low uncertainty participate in the risky asset
markets.7 As a result, there are two forces affecting the equity premium when market participation is reduced. On one hand, since fewer investors hold the risky assets, the risk premium
is higher. On the other hand, because the investors who participate in the markets have lower
model uncertainty, the uncertainty premium is lower. The effect of limited market participation on risk premium has been documented in the literature (Mankiw and Zeldes (1991),
Basak and Cuoco (1998), Vissing-Jorgensen (1999), and Brav, Constantinides, and Geczy
(2002)). However, these studies do not consider model uncertainty or uncertainty premium.
In our model, the uncertainty premium is lower in the economy in which there is limited
market participation. The decrease in the uncertainty premium outweighs the increase in
the risk premium, leading to a lower equity premium in the economy in which limited market
participation occurs than that in which full participation takes place. Furthermore, the equity premium decreases as the model uncertainty becomes increasingly dispersed and more
investors stay sidelined.
Basak and Cuoco (1998) study the implications of exogenous limited market participation
on asset prices. While similar in focus, the basic ideas behind our respective models are quite
different. In the Basak and Cuoco model, some investors (with log utility) are assumed not
to invest in stocks, while other investors (with CRRA utility) are assumed to trade both
the stocks and the riskless asset. In equilibrium, the unrestricted investors have to hold
all the stocks and bear all the market risk. Moreover, this group of investors have to lever
their equity positions by borrowing at the risk free rate, because the restricted investors can
hold only the risk free asset. For this to happen, the equilibrium risk free rate has to be
lower, leading to a larger equity premium. In our model, limited market participation arises
7
In standard models, risk aversion alone cannot explain why investors have zero rather than small holdings
in risky assets.
4
endogenously and, in sharp contrast to Basak and Cuoco (1998), equity premium is lower as
market participation rate decreases.
The intuition for diversification discount is also straightforward. When there is limited
market participation, the pre-merger price of each of the two firms is determined by investors
with less model uncertainty for that firm. They therefore demand a lower uncertainty premium and are willing to pay higher prices for the risky assets. After the merger investors
have to buy both firms as a bundle. An investor who is less uncertain about one firm is not
necessarily less uncertain about the other firm and is only willing to offer a lower price for
the combined firm than the sum of prices of the two firms trading separately. Thus, when
there is limited market participation, a merger will result in a decrease in the price of the
combined firm. Our paper provides a new explanation for the conglomerate discount and
spin-off premium phenomena based on investors’ model uncertainty.
Our paper has the following empirical implications. When using the dispersion of analysts’ forecasts as a proxy for the model uncertainty dispersion among investors, stock
returns should be negatively related to this proxy. This is supported by a recent study of
Diether, Malloy and Scherbina (2002) on the effect of the dispersion in analysts’ forecasts on
stock returns.8 Our model predicts that limited participation is associated with lower equity
premium. This is consistent with Chen, Hong and Stein (2002), who find that a narrower
breath of ownership is related to lower expected stock returns.9 With respect to the diversification discount, our model predicts that diversification discount and spin-off premium will
be larger when investors are more heterogeneous. It would be interesting to determine how
breadth of ownership, dispersion in analysts’ forecasts are related to diversification discount
and spin-off premium.10
The rest of the paper is organized as follows. Section 2 describes the model. Section
3 discusses investors’ portfolio choices and their market participation decisions. Section 4
discusses how the limited market participation relates to the equity premium, while Section
8
While these authors state that their results support Miller’s (1977) model based on heterogeneous beliefs
in the presence of short-sale constraint, the empricial findings are also consistent with our prediction which
is based on model uncertainty in financial markets. Moreover, our model also predicts that returns are
negatively correlated with dispersions in analysts’ forecasts even in markets without short-sale constraint.
9
Interestingly, Chen, Hong and Stein (2002) also interpret their empirical results with a Miller-type model
based on heterogeneous beliefs and short-sale constraint.
10
Other proxies of investor heterogeneity may include short interest and trading volume.
5
5 investigates the relation between the diversification discount and limited market participation. In Section 6, we discuss the robustness of our results in a more general setting
with richer factor structure for asset payoffs including the idiosyncratic factors. Section 7
concludes. Proofs and some technical details are collected in the appendix.
2
The Model
To highlight the intuition, we make several simplifying assumptions about the payoffs of the
risky assets. Some of these assumptions are relaxed in Section 6 in which we show that the
qualitative results obtained under the simplifying assumptions also hold in a more general
setting.
2.1
The Setting
We assume a one-period heterogeneous agent economy. Investment decisions are made at
the beginning of the period while consumption takes place at the end of the period. There
are two risky assets and one risk-free asset in the economy. The risk-free asset is in zero net
supply and the risk-free rate is set at zero. The payoffs on the two risky assets follow a two
factor model
r1 = fA + ρfB ,
(1)
r2 = ρfA + fB ,
(2)
where r1 and r2 are payoffs on asset 1 and 2, respectively, fA and fB are two (random)
systematic factors, and ρ represents the factor loading. We assume |ρ| < 1. Agents are
endowed with shares of the risky assets. The total supplies of the two risky assets are given
by x1 and x2 , with x1 = x2 = x > 0. All variables are jointly normally distributed. Investors
do not have perfect knowledge of the distribution of the random factors in the economy. More
specifically, they know that the factors, fA and fB , follow a joint normal distribution. The
risk of asset payoffs is summarized by the non-degenerate variance-covariance matrix ΩF . We
assume that investors have precise knowledge of ΩF . Without loss of generality, we assume
that ΩF is diagonal with common diagonal element σf . However, investors do not know
6
exactly the mean of fA or fB . This is motivated by the fact that it is much easier to obtain
accurate estimates of the variance and covariance of the state variables than their expected
values, e.g., Merton (1992). The imperfect knowledge of the asset payoff distribution gives
rise to model uncertainty.
2.2
The Preferences
Each agent in the economy has a CARA utility function u(W ). Due to the lack of perfect
knowledge of the probability law of asset payoffs, the agent’s preference is represented by a
multi-prior expected utility (Gilboa and Schmeidler (1989))
min EQ [u(W )] ,
Q∈P
(3)
where EQ denotes the expectation under the probability measure Q, P is a set of probability
measures. The general nature and the axiomatic foundation of these preferences has been
well studied in the literature (Gilboa and Schmeidler (1989)). The basic intuition behind
this preference is that in the real world, people do not know precisely the true probability
distribution of asset payoffs. They may have a reference model of the probability distribution,
such as a normal distribution with certain mean and variance. The reference model could
be the result of some econometric analysis. However, they are not completely sure of the
correctness of the reference model, for example their estimates of the mean payoffs of the
assets. As a result, they are willing to accept a set P of probability distributions to which
the true probability distribution/model belongs. Since these investors are averse to (model)
uncertainty, they evaluate the expected utility of a random consumption, E Q [u(W )] under
the worst case scenario.
The specification of the set P follows Kogan and Wang (2002) closely. In the interest of
focusing on the issues we study, we provide a brief description of P and its intuition. More
details can be found in Kogan and Wang (2002). Let µA = µB = µ be the reference level
for the mean of fA and fB . They are typically the result of econometric estimation. We
will assume that all investors have the same point estimate µ, but their uncertainty about
the estimate differs.11 This could be due to heterogeneity among the investors regarding the
11
Our results regarding limited market participation, the equity premium, and the diversification discount
7
precision of the point estimate. Let vA and vB be the adjustment to the mean µA and µB ,
respectively. We therefore define the set P by the following constraints on vA and vB .12
vA2 σf−2 ≤ φ2A ,
(4)
vB2 σf−2 ≤ φ2B ,
(5)
and
where φA and φB are parameters that capture the investor’s uncertainty about the mean of
factor A and B, respectively. In other words, investors believe the true mean of factor fA
and fB fall into the set
{(µA + vA , µB + vB ) : vA and vB satisfy (4) and (5), respectively}.
Note that the higher the φA (φB ), the wider the range for the expected payoff of fA (fB ).
Thus higher φA (φB ) means higher uncertainty about fA (fB ). So we call φA (φB ) the level
of uncertainty of the investor about the mean of factor A (B).
In our economy, investors have the same risk aversion, but are heterogeneous in their
level of model uncertainty. We assume that φ = (φA , φB ) are uniformly distributed on the
square13
S = [φ̄ − σφ , φ̄ + σφ ] × [φ̄ − σφ , φ̄ + σφ ],
where φ̄ ≥ σφ . When φ̄ = σφ = 0, there is no model uncertainty and our model collapses to
the standard expected utility model.
3
Equilibrium Market Participation
In this section, we find the closed-form solution for the equilibrium and show that when
investors’ heterogeneity in model uncertainty is high, there is limited market participation
hold without the assumption of the same estimate.
12
As Kogan and Wang (2002) show, any alternative measure Q is represented by the adjustments to the
mean of µA and µB , respectively. Moreover, if the set P is log-likelihood ratio based, then under normal
distribution assumption, the set P is always characterized by quadratic inequalities in the adjustment to the
mean.
13
More general distributions are considered later.
8
in equilibrium.
3.1
Portfolio Choices
We first analyze individual investor i’s portfolio choice problem. The investor’s wealth
constraint is
W1i = W0i + Di1 (r1 − P1 ) + Di2 (r2 − P2 ),
where Di1 , Di2 denote investor i’s demand for the two risky assets. Due to the assumption
of normality and the CARA utility, investor i’s utility maximization problem becomes:
max min
Di1 ,Di2 viA ,viB
−Di1 P1 − Di2 P2 + (Di1 + ρDi2 )(µ − viA ) + (Di2 + ρDi1 )(µ − viB )
−
γσf2
2
[(Di1 + ρDi2 )2 + (Di2 + ρDi1 )2 ] ,
where γ is the investor’s risk aversion coefficient. The first two terms of the objective function
represent the prices that investors pay for the risky assets; the next two terms represent the
expected payoff from exposure to factors A and B with adjustment to the mean due to
model uncertainty; the last two terms represent adjustment due to risk. Solving the inner
minimization, the above problem reduces to
maxDi1 ,Di2 −Di1 P1 − Di2 P2 + (Di1 + ρDi2 )[µ − sgn(Di1 + ρDi2 )φiA σf ]
+(Di2 + ρDi1 )[µ − sgn(Di2 + ρDi1 )φiB σf ] −
γσf2
2
[(Di1 + ρDi2 )2 + (Di2 + ρDi1 )2 ] ,
where sgn(·) is an indicator function which takes the sign of its argument.
It will be convenient to introduce factor portfolios tracking the two factors A and B. Let
DiA and DiB be investor i’s holdings of the factor portfolios A and B, respectively. Given
the structure for asset payoffs, we have investor i’s demand for the factor portfolios A and
B given by:
DiA ≡ Di1 + ρDi2 ,
DiB ≡ ρDi1 + Di2 .
9
Denote by PA the price of factor portfolio A. Due to symmetry, the prices of the two
factors are the same. The investor’s optimal holding in factor portfolio A is then given by



DiA =


1
(µ
γσf2
0
1
(µ
γσf2
− φiA σf − PA ) if µ − PA > φiA σf ,
if − φiA σf ≤ µ − PA ≤ φiA σf ,
+ φiA σf − PA ) if µ − PA < −φiA σf .
(6)
The demand function for factor portfolio B, DiB , is analogous by symmetry. It can be seen
from (6) that, due to his uncertainty about the true mean of factor A, investor i’s trading
strategy is very conservative. He buys (sells short) factor portfolio A only when the equity
premium is strictly positive (negative) in the worst case scenario, vA = φiA σf (vA = −φiA σf ).
When the price PA falls in the range [µ − φiA σf , µ + φiA σf ], the investor will not participate
in the market for factor portfolio A. This is in sharp contrast to the standard result based
on expected utility model that as long as the equity premium is positive (µ − PA > 0), the
investor will hold the risky factor portfolio. Given the optimal demand for factor portfolios,
the optimal demand for the two risky assets can be obtained using the linear relation between
the demand for the factor portfolios and the demand for the risky assets,
3.2
Di1
Di2
1
=
1 − ρ2
1 −ρ
−ρ 1
DiA
DiB
.
(7)
Equilibrium with Full Participation
In equilibrium, the market clears. The supply for each factor portfolio, x̂ = (1 + ρ)x, is equal
to its aggregate demand. Due to the symmetry of asset payoffs, the solutions for the demand
and price of the two factor portfolios are analogous. We therefore focus on factor portfolio
A. As will be seen later, in equilibrium, the factor portfolio always sells at a discount,
i.e., µ − PA > 0. Given this, the expression of the demand function, Equation (6), implies
that, ignoring the initial endowment, no investor will sell short any factor portfolio. In the
full participation equilibrium, all investors participate in the asset markets. So the market
clearing condition for factor portfolio A can be written as
x̂ =
1
1
1
(µ − σf φiA − PA ) 2 dφiA dφiB =
(µ − σf φ̄ − PA ).
2
4σφ
γσf2
φi ∈S γσf
10
Solving for PA , we arrive at the following equilibrium price for the factor portfolio A,
PA = µ − x̂γσf2 − σf φ̄.
(8)
The first two terms of this expression are standard. They represent the expected payoff
and the risk premium, respectively. The third term is new. It represents the uncertainty
premium. Note that φ̄ is the average level of uncertainty of all investors about the mean
payoff of the factor portfolio.
The full participation equilibrium prevails when all investors participate in the asset
markets. This requires almost all investors (including investors with the highest uncertainty)
hold long positions. Equation (6) thus implies that the following condition must be satisfied:
µ − (φ̄ + σφ )σf − PA > 0.
Substituting (8) into the expression, we have that in equilibrium all investors participate in
the market if
x̂γσf > σφ .
(9)
This full participation condition indicates that whether all investors participate in the markets depends on the dispersion of investors’ model uncertainty. When σφ is small, investors
are reasonably homogeneous and all investors will participate. It also depends on the risk
premium, x̂γσf2 . Given the model uncertainty, the higher the risk premium, the higher the
total equilibrium equity premium, µ − PA , and the more likely investors with the highest
uncertainty will participate (Equation (6)).
It is interesting that under full market participation, investors’ model uncertainty dispersion has no effect on price. What matters is the average model uncertainty. Investors
with more uncertainty will hold less risky assets than the average investor while investors
with less uncertainty will hold more. When all investors fully participate, the market price
behaves as if all investors have the average model uncertainty.
3.3
Limited Participation
When condition (9) is not satisfied, investors with high uncertainty will not participate. Let
φ∗ denote the lowest level of uncertainty at which investors do not hold factor portfolio A.
11
Then
µ − σf φ∗ − PA = 0.
(10)
Investors whose uncertainty about factor A is higher than φ∗ will choose not to participate
in factor portfilio A.
The market clearing condition is given by
x̂ =
φiA <φ∗
DiA
1 φ∗ − φ̄ + σφ
σf (φ∗ + φ̄ − σφ )
dφiA =
[µ
−
− PA ].
2σφ
γσf2
2σφ
2
Solving for φ∗ yields
φ∗ = φ̄ − σφ + 2
γσφ σf x̂.
(11)
(12)
Using Equation (10), we arrive at the following equilibrium price for the factor portfolio A:
PA = µ − σf (φ̄ − σφ ) − 2σf
γ x̂σφ σf .
(13)
The prices of the original assets can be obtained as follows
P1 = P2 = PA (1 + ρ).
To gain intuition of the pricing formula, we rewrite Equation (11) as
µ − PA =
γσf2 x̂
+ φA σf ,
αA
where
φ∗ − (φ̄ − σφ )
αA =
=
2σφ
(14)
γσf x̂
σφ
is the proportion of investors who hold factor portfolio A, and
1
φA = (φ∗ + φ̄ − σφ ) = φ̄ − σφ + γσφ σf x̂
2
can be interpreted as the average level of uncertainty for factor portfolio A among the market
participants. Equation (14) suggests that the total equity premium can be decomposed into
12
two components. The first component is the risk premium and the second component is
the uncertainty premium. The lower the participation rate, the higher the risk premium.
This is consistent with what is known in the literature (for example, Basak and Cuoco
(1998)). However, both the participation rate and the uncertainty premium decrease in the
uncertainty dispersion (σφ ). Consequently, the lower uncertainty premium is associated with
the lower participation rate.
Figure 1 provides some simple comparative static results with respect to the average
model uncertainty (φ̄), supply of the asset (x), risk aversion (γ), and the model uncertainty
dispersion (σφ ). The price decreases in the average model uncertainty and the supply. Given
the asset payoff, higher model uncertainty and supply make the asset less attractive to
investors, which leads to a lower price. The equilibrium price also decreases in investors’ risk
aversion. Intuitively, investors with low risk aversion are more likely to buy a risky asset
for the same future payoff than investors with high risk aversion. As investors’ risk aversion
decreases, the demand for risky assets increases. This leads to higher prices for the risky
assets. As the model uncertainty dispersion increases, investors who are more uncertain
about asset mean payoffs gradually withdraw from the markets, leaving those who are less
uncertain in the markets. Because these remaining investors are willing to pay higher price
for the assets, the price increases. Note that when the model uncertainty dispersion is low,
the risky asset price does not vary with the uncertainty dispersion. This is because there
is full market participation and the equilibrium price is determined by the average model
uncertainty and not affected by the dispersion of model uncertainty.
In Figure 2 we plot the percentage of nonparticipation in risky asset markets, (1 − αA )2 ,
as a function of asset payoff volatility (σf ), asset supply (x), risk aversion (γ), and the
model uncertainty dispersion (σφ ). The percentage of investors not participating in the risky
asset markets decreases as the asset payoff volatility, supply of the asset, and investors’ risk
aversion increase. This is caused by the lower price associated with higher volatility, supply
of the asset, and risk aversion. On the other hand, because the price is higher at higher
dispersion of model uncertainty, investors are increasingly staying sidelined as the model
uncertainty dispersion increase. This finding is in contrast with the result in the standard
expected utility model where investors will always take some position in the asset when there
is a positive risk premium.
13
4
Market Participation and Equity Premium
There is a growing literature on limited market participation. A widely investigated issue
in this literature is how the equity premium relates to limited market participation. It
is found that the equity premium increases as fewer investors participate in the markets
(Mankiw and Zeldes (1991), Basak and Cuoco (1998), Vissing-Jorgensen (1999), and Brav,
Constantinides, and Geczy (2002)). However, these studies assume that some investors are
exogenously excluded from participating in certain markets.
We re-examine the relation between the equity premium and limited market participation
by allowing investors to optimally choose whether or not to participate in the markets.
Therefore, limited market participation arises endogenously in our economy. We next state
the main result of this section.
Proposition 1 If x̂γσf < σφ , then there is limited market participation in equilibrium; and
furthermore (a) investors’ participation rate decreases with the uncertainty dispersion, i.e.,
∂αA /∂σφ < 0; (b) the average model uncertainty for factor portfolios decreases with the
uncertainty dispersion, i.e., ∂φA /∂σφ < 0; and (c) the equity premium decreases with the
uncertainty dispersion for both the factor portfolios (∂(µ − PA )/∂σφ < 0) and the assets
(∂(µj − Pj )/∂σφ < 0, where µj = (1 + ρ)µ for j = 1, 2).
Proposition 1 indicates that at high model uncertainty dispersion some investors will not
participate in the risky asset markets in equilibrium. In particular, investors with uncertainty
much higher than the average uncertainty optimally choose to stay sidelined. In this case,
for these investors to take long positions in the factor portfolios, the expected uncertaintyadjusted payoff is too low. Similarly, for these investors to take short positions, the expected
uncertainty-adjusted payoff is too high.
We have shown earlier that with full market participation, the equity premium does not
depend upon model uncertainty dispersion among investors. However, with limited market
participation, Proposition 1 states that the equity premium decreases with model uncertainty
dispersion. The intuition is readily seen from Equation (14). As the uncertainty dispersion
14
increases, according to Proposition 1(a), market participation rate decreases and risk premium increases. At the same time, however, uncertainty premium decreases according to
Proposition 1(b). This is because the remaining market participants have lower uncertainty
about the mean payoffs of the risky assets and are willing to accept lower uncertainty premium. The second effect dominates the first leading to a lower equity premium as shown
in Proposition 1(c). This is in sharp contrast with the result in the existing limited market
participation literature.
The following proposition provides some further results regarding the comparative statics of the uncertainty premium and the risk premium with respect to the volatility of the
risky assets, the investors’ risk aversion, average model uncertainty, and model uncertainty
dispersion.
Proposition 2 Both the uncertainty premium and the risk premium increase with the variance of the risky asset payoff (σf ) and the risk aversion coefficient (γ). The risk premium
is unrelated to the average model uncertainty (φ̄) but increases with the dispersion of model
uncertainty (σφ ). However, the uncertainty premium increases with the average model uncertainty (φ̄) but decreases with the dispersion of model uncertainty (σφ ).
To gauge the relative importance of the two components of the equity premium, we conduct some numerical analysis. Figure 3 shows the equity premium (solid line) decomposed
into the uncertainty premium (dashed line) and the risk premium (dash-dotted line) as a
function of the supply of the asset (x), the payoff volatility (σf ), the investor’s risk aversion
(γ), and the model uncertainty dispersion among investors (σφ ). Both the uncertainty premium and the risk premium as well as the total equity premium are increasing in the supply
of the asset and the volatility of asset payoff. This reflects the fact that investors demand
higher returns to hold more risky assets and to compensate them for higher risk. Similarly,
as investors’ risk aversion increases, a higher risk premium is required to induce investors to
hold the risky assets. Furthermore, a higher risk aversion increases investors’ average level
of uncertainty, hence the uncertainty premium. When the risk aversion is sufficiently high,
full market participation prevails and the uncertainty premium stays constant while the risk
premium keeps increasing. Consequently, the total equity premium increases as investors
15
become more averse to risk.
Interestingly, while the risk premium increases with investors’ model uncertainty dispersion, the uncertainty premium initially stays flat and then decreases as the model uncertainty
becomes more dispersed among investors. This happens when there is full market participation at low levels of uncertainty dispersion. At full market participation with low model
uncertainty dispersion, the uncertainty premium only depends on the average level of model
uncertainty and is not affected by the model uncertainty dispersion. As the model uncertainty dispersion increases, investors who have high aversion to model uncertainty stay
sidelined, which creates limited market participation. When limited market participation
takes place, the uncertainty premium decreases with the dispersion of model uncertainty.
In fact, the decrease in uncertainty premium outweighs the increase in the risk premium so
that the total equity premium also decreases. Intuitively, as the model uncertainty becomes
more dispersed, investors with very high uncertainty aversion will be sidelined, but investors
with very low uncertainty aversion are relatively optimistic and stay in the market. The
equilibrium price is now determined by more optimistic investors. While the optimistic investors’ risk premium increases because fewer investors stay in the market, their uncertainty
premium decreases. Since the uncertainty premium dominates the risk premium, the equity
premium is reduced. Our finding is in sharp contrast to the result of Basak and Cuoco (1998)
who take market participation decision as exogenously given and show that equity premium
increases when there is more restricted participation. However, we have shown here that
when market participation is endogenously determined, lower market participation is not
necessarily associated with higher equity premium.
5
Market Participation and Diversification Discount
We have shown that limited market participation can occur when uncertainty dispersion is
sufficiently large among investors. It will be interesting to explore how market participation
will change when the markets are made more incomplete through mergers and acquisitions.
We now consider the case in which two firms are combined to form one conglomerate. We
examine how, in the absence of any synergy between the two firms, the market value of the
conglomerate depends upon investors’ model uncertainty dispersion and how it relates to
16
market participation.
We assume that the firms combine their operations and cash flows to form a conglomerate
firm denoted M. Furthermore, the cash flows are unaffected by the merger.14 The payoff
from firm M is rM = r1 + r2 = (1 + ρ)(fA + fB ). Investors can no longer trade in asset 1
and 2 and can only hold asset M.
5.1
Full Participation
We first consider the case of full market participation. Let DiM be investor i’s demand for
the conglomerate and PM be the market price of the conglomerate. Investor i’s optimization
problem is:
2
max DiM [2(1 + ρ)µ − (1 + ρ)sgn(DiM )(φiA + φiB )σf − PM ] − γDiM
(1 + ρ)2 σf2 .
DiM
Combining the solution with the market clearing condition, we arrive at the following proposition on the price of the conglomerate.
Proposition 3 In a full participation equilibrium, i.e., x̂γσf > σφ , the price for the conglomerate is given by
PM = 2(1 + ρ)[µ − σf φ̄ − γ x̂σf2 ].
(15)
Equation (15) indicates that the price of the conglomerate is the simple sum of the
market value of the two firms trading separately, i.e., PM = P1 + P2 . Investors who would
have preferred to buy more of factor portfolio A than portfolio B now buy both portfolios
with equal weight. Similarly, investors who would like to buy more of portfolio B than
portfolio A now buy both at equal weight. However, because of full market participation,
the price is still determined by the average investor who has model uncertainty φ̄ for both
factors A and B. With or without merger, the average investor will be holding the market
portfolio. As a result, there is no diversification discount.
14
We abstract from agency problems such as inefficient internal capital markets and rent-seeking at
division-level considered by Rajan, Servaes and Zingales (2000) and Scharfstein and Stein (2000).
17
5.2
Limited Participation
Next, we consider the case of limited market participation. The result is summarized in the
following proposition.
Proposition 4 If x̂γσf < σφ , i.e., if there is limited market participation in equilibrium,
then
PM = 2(1 + ρ)[µ − σf g −1 (γ x̂σf )/2],
where
g(y) = (y − 2φ̄ + 2σφ )3 − 2[(y − 2φ̄)+ ]3 /{48σφ2 },
and PM < P1 + P2 , i.e., there is a diversification discount.
This proposition states that, with limited market participation, the price of a conglomerate is no longer equal to the sum of its component prices when traded separately. The
reason is that prices for the factor portfolios are determined by investors with low uncertainty. When the firms are merged, an investor who has low uncertainty on factor A does
not necessarily have low uncertainty on factor B. If there is no diversification discount, i.e.,
PM = P1 + P2 , there will not be sufficient demand for the conglomerate. Thus, when the
two assets are selling as a bundle only, the price has to be reduced to attract investors.
The proposition predicts that there exists a conglomerate discount when there is limited
market participation. This is supported by the empirical findings of a diversification discount
for conglomerates documented in Lang and Stulz (1992), Berger and Ofek (1995), and Rajan,
Servaes, Zingales (2000). Moreover, our model is also consistent with a spin-off premium
demonstrated by Hite and Owers (1983), Miles and Rosenfeld (1983) and Schipper and Smith
(1983) using event studies.
The following two corollaries provide further characterization of the relation between the
conglomerate discount on one hand and the model uncertainty, market participation on the
other.
Corollary 1 When there is limited market participation, diversification discount increases
with model uncertainty dispersion (σφ ).
18
In terms of its empirical implication, Corollary 1 suggests that if we can use some proxies
for investors’ model uncertainty dispersion, the diversification discount should be positively
correlated with these proxies.
In Figure 4 we plot the diversification discount as a function of the model uncertainty
dispersion among investors (σφ ). Both the discounts measured in level and in percentage
show the same features. Initially, at low levels of the model uncertainty dispersion, there is
full market participation and the diversification discount is zero. As the uncertainty dispersion becomes sufficiently large, we have limited market participation and the diversification
discount becomes positive. With limited market participation, as the uncertainty dispersion
increases and more investors stay sidelined, the diversification discount also increases. Both
panels in the figure suggest that the discount can be quite sizable.
Corollary 2 When there is limited market participation, a conglomerate merger will further
reduce market participation.
Figure 5 plots the nonparticipation rate for both the case with separate trading of two
assets (solid line) and the case in which the two assets merged into one conglomerate (dashed
line). The figure illustrates that at low levels of model uncertainty dispersion, we observe
full market participation. When the model uncertainty dispersion becomes sufficiently large,
some investors optimally choose to stay sidelined. As the model uncertainty further increases,
the fraction of investors withdrawn from the market increases. Moreover, the percentage of
investors staying sidelined is always higher when investors can only trade the conglomerate
than when they can trade both assets separately. Intuitively, when two assets are traded
separately, investors who have high value on one asset can own this particular asset only.
When two assets are combined and traded as a bundle, these same investors may withdraw
from the market because the benefit of owning this asset is outweighed by the cost of having
to simultaneously own the other asset for which they have a low valuation. Consequently,
more investors will choose to stay away from the risky asset markets.
19
6
Generalization
In this section we extend the model in Section 2 to allow for a more general factor structure.
Specifically, the payoffs of risky asset 1 and 2 are now given by
r1 = β1A fA + β1B fB + 1 ,
(16)
r2 = β2A fA + β2B fB + 2 ,
(17)
where βkj , k = 1, 2, j = A, B, are factor loadings and invertable, fA and fB are the
systematic factors as before, and 1 and 2 are the idiosyncratic factor for asset 1 and 2,
respectively.
All variables are jointly normally distributed. The idiosyncratic factors are uncorrelated
with each other and are uncorrelated with the systematic factors. As in Section 2, investors
do not have perfect knowledge of the distribution of the random variables in the economy.
They know that the factors (fA , fB , 1 , 2 ) follow a joint normal distribution. The risk
of asset payoffs is summarized by the non-degenerate variance-covariance matrix ΩF . We
assume that investors have precise knowledge of ΩF . Without loss of generality, we assume
that fA and fB have equal variance σf2 . However, investors do not know exactly the mean
of fA and fB .15 Investors’ uncertainty is represented by the constraints,
2
viA
σf−2 ≤ φ2iA ,
2
viB
σf−2 ≤ φ2iB .
Similar to Section 2, investors’ heterogeneity is characterized by φi = (φiA , φiB ). We assume
that φiA and φiB have a joint distribution defined on [φlA , φuA ] × [φlB , φuB ] with a continuous
probability density function h(φiA , φiB ), where φlj and φuj , j = A, B, are the lower and
upper bound, respectively. The average of φi is denoted by φ̄ = (φ̄A , φ̄B ) .
15
By assumption, the idiosyncratic factors have zero mean.
20
6.1
Portfolio Choice and Market Participation
Due to the assumption of normality and the CARA utility, investor i’s maximization problem
is reduced to:
max min −Di1 P1 − Di2 P2 + (β1A Di1 + β2A Di2 )(µA − viA ) + (β1B Di1 + β2B Di2 )(µB − viB )
Di1 ,Di2 viA ,viB
−
γ
2 2
2 2
(β1A Di1 + β2A Di2 )2 σf2 + (β1B Di1 + β2B Di2 )2 σf2 + Di1
σ1 + Di2
σ2 .
2
(18)
The factor portfolios for fA and fB are given by:
DiA ≡ β1A Di1 + β2A Di2 ,
(19)
DiB ≡ β1B Di1 + β2B Di2 .
(20)
Since the matrix
β=
β1A β2A
β1B β2B
,
is invertible, there is an one-to-one correspondence between the holdings of factor portfolios
and those of the asset 1 and 2. In particular, if an investor does not participate in the factor
portfolio markets, he will not participate in the market of asset 1 or 2.
The first order condition for investor i’s optimal holdings in factor portfolios is given by
µA − φiA σf + λiA
µB − φiB σf + λiB
−P =γ
σf2 0
0 σf2
−1 + (β )
σ12 0
0 σ22
β
−1
DiA
DiB
,
where λiA and λiB are nonnegative numbers such that 0 ≤ λiA ≤ 2σf φiA and 0 ≤ λiB ≤
2σf φiB , and P ≡ (PA , PB ) = (β −1 ) (P1 , P2 ) . These constraints imply that when the
investor holds a long position in factor A, the first order derivative with respect to DiA is
positive and investor i takes the lowest expectation among his priors. Similarly, if investor i
holds a negative position, the first order condition holds and he takes the highest expectation
among his priors and thus λiA = 2σf φiA . When he does not hold any position in factor A,
the first order derivative changes sign from positive to negative around DiA = 0. In this case
0 ≤ λiA ≤ 2σf φiA . Similar arguments holds for the constraint on λiB .
21
Let Di = (DiA , DiB ) , λi = (λiA , λiB ) , it follows that the demand for factor portfolios
is
Di = (γ Ω̂)−1 (µ − φi σf + λi − P ) ,
where
Ω̂ =
σf2 0
0 σf2
+ (β
−1 )
σ12 0
0 σ22
(21)
β −1
is the variance-covariance matrix of the factor portfolios. Without loss of generality, we
assume that Ω̂ is positive.16 The demand for factor portfolios can be rewritten as
Di = (γ Ω̂)−1 [µ − P ] − (γ Ω̂)−1 [φi σf − λi ] .
(22)
The first term on the right hand side is the standard mean-variance demand, and the second term is the adjustment to the mean-variance demand due to model uncertainty. We
summarize the result on market participation in the general setting as follows.
Theorem 1 If min{φuA , φuB } > M ≡ max{|µA − PA |, |µB − PB |}, then investors with
uncertainty level in S = [M, φuA ] × [M, φuB ] will take no positions in the risky asset markets.
Investors with large uncertainty about payoffs stay away from the financial market. There
is too much uncertainty for them to hold either long or short positions.
6.2
Equilibrium and Market Participation
In the general setting of this section, we do not have a closed-form solution for the equilibrium. However, the existence of equilibrium is guaranteed by the following theorem.
Theorem 2 There exists an equilibrium. In addition, if the supply of both factor portfolios
are nonzero, the equilibrium is unique.
Note that when the supply of one of the factor portfolios is zero, there could exist multiple
prices for that asset. Multi-equilibria occur because there could exist a continuum of prices
16
Otherwise, we can redefine a new set of factors fA = −fA , fB = fB such that Ω̂ is positive.
22
for that factor portfolio such that investors choose not to participate at these prices and
demand equals supply as a result.
6.3
Equity Premium under Limited Participation
Let x1 and x2 be the supply of asset 1 and 2, respectively. The corresponding factor portfolio
supply is given by
x≡
xA
xB
=β
x1
x2
.
We will assume now that xA > 0 and xB > 0.
In equilibrium, the asset markets must clear, or equivalently, the factor portfolio markets
must clear, i.e.,
xA =
xB =
DiA =0
DiB =0
DiA h(φiA , φiB )dφiA dφiB ,
(23)
DiB h(φiA , φiB )dφiA dφiB .
(24)
We say that there is limited participation in factor portfolio j, j = A, B when the
measure of nonparticipants in factor portfolio j is positive. We have the following results
regarding the relation between limited participation and asset prices.
Theorem 3 Let ωij denote the (i, j)th component of Ω̂−1 .
(a) If γx ≥ σf σφ (ω11 − ω12 , ω22 − ω12 ) , then all investors hold both factor A and factor B
portfolios; and the full market participation equilibrium prices for the factor portfolios,
P f ≡ (PAf , PBf ) , are given by
P f = µ − σf φ̄ − Ω̂x;
(25)
(b) If there is limited participation in factor portfolio A (B) but full participation in factor
portfolio B (A), then PA > PAf and PB = PBf (PA = PAf and PB > PBf );
23
(c) If there is limited market participation in both factor portfolios, PA > PAf and PB > PBf .
Market participation clearly depends on how dispersed investors’ model uncertainty is.
When investors are relatively homogeneous, φuA − φlA and φuB − φlB are small, a full participation equilibrium will prevail. When investors’ model uncertainty is relatively homogeneous
with respect to one factor but not the other, there could be limited participation with respect
to one of the factor portfolios. Finally, when investors’ model uncertainty is very heterogeneous with respect to both factors, some investors will not participate in either risky asset.
6.4
Diversification Discount
In this section, we show that our results on the diversification discount still go through in
the general setting.
Theorem 4 There is diversification discount, i.e., PM < x P = x1 P1 + x2 P2 , if and only
if there is limited market participation in equilibrium.
Intuitively, when there is limited market participation, some investors prefer staying
away from one of the factor portfolios but not the other. Bundling of assets forces some
investors to take on portfolios they would rather avoid and this leads to a diversification
discount.
7
Conclusion
We investigate limited market participation and its relation to the equity premium and the
diversification discount in an equilibrium framework with heterogeneous model uncertainty.
We find that sufficiently large dispersion in model uncertainty among investors leads to
limited market participation. Equity premium in our model can be decomposed into two
components: the uncertainty premium and risk premium. The uncertainty premium increases in the average model uncertainty and decreases in the uncertainty dispersion among
investors, whereas the risk premium can be affected by the uncertainty dispersion but not
24
the average model uncertainty. When there is full market participation, the uncertainty
premium is positively related to the average model uncertainty but unrelated to the model
uncertainty dispersion. The risk premium is unaffected by the uncertainty dispersion. Overall, the equity premium increases in average model uncertainty but is not affected by the
model uncertainty dispersion. When there is limited market participation, the risk premium
increases. However, the uncertainty premium decreases as the model uncertainty dispersion
increases. The lower uncertainty premium outweighs the higher risk premium. The equity
premium therefore decreases as the model uncertainty dispersion increases. Because market
participation rate decreases with the model uncertainty dispersion, the equity premium is
negatively related to market participation rate. This is in sharp contrast to the existing
understanding that limited market participation helps to resolve the equity premium puzzle.
Interestingly, we show that the diversification discount puzzle is related to the limited
market participation puzzle. When there is limited market participation, a conglomerate
merger can result in a price discount compared to single segment firms (Conversely, spin-off
of a conglomerate can result in a spin-off premium). Both diversification discount and limited
market participation are caused by model uncertainty dispersion among investors. The
diversification discount increases as more investors stay sidelined. Moreover, a conglomerate
merger will decrease market participation. Our study provides a new explanation for the
diversification discount puzzle from the asset pricing perspective.
25
Appendix
Proof of Proposition 1 and Proposition 2: The propositions follow readily from equations (12) and (14), the expression for the market participation rate αA , the expression of
market participants’ average uncertainty φA , and the relation Pj = PA (1 + ρ), j = 1, 2.17
Proof of Proposition 3 and Proposition 4: When γ x̂σf ≤ σφ , we have
DiM =
2(1 + ρ)µ − (1 + ρ)(φiA + φiB )σf − PM
.
2(1 + ρ)2 γσf2
Aggregating the demands across all investors and equal the aggregate demand to aggregate
supply, we arrive at
2(1 + ρ)µ − 2(1 + ρ)φ̄σf − PM
.
2(1 + ρ)2 γσf2
x=
Consequently, we have
PM = 2(1 + ρ)[µ − φ̄ − γ x̂σf2 ] = (1 + ρ)(PA + PB ) = P1 + P2 ,
and there is no diversification discount.
When γ x̂σf < σφ , there exists a φ̂ such that when φiA + φiB ≥ φ̂, investor i will not
invest in the conglomerate, i.e.,
[2(1 + ρ)µ − (1 + ρ)φ̂σf − PM ] = 0.
(A1)
Notice that no investor will sell short the conglomerate and thus DiM > 0 for investors participating in the market. Consequently, sgn(DiM ) = 1 for participating investors. Summing
up investors’ demand, we have
x=
φiA +φiB <φ̂
1
[2(1 + ρ)µ − (1 + ρ)(φiA + φiB )σf − PM ]dφiA dφiB . (A2)
8(1 + ρ)2 γσf2 σφ2
Utilizing Equation (A1), we arrive at the following market clearing condition
x=
φiA +φiB <φ̂
17
[φ̂ − (φiA + φiB )]
dφiA dφiB .
8(1 + ρ)γσf σφ2
Note that in the particular economy we are studying, αA = αB due to symmetry.
26
(A3)
Notice that
φiA +φiB <φ̂
(φ̂ − φiA − φiB )
(φ̂ − 2φ̄ + 2σφ )3 − 2[(φ̂ − 2φ̄)+ ]3
dφ
dφ
=
.
iA
iB
8σφ2
48σφ2
Define g(φ̂) as the right hand side of the above equation, i.e.,
g(φ̂) ≡
(φ̂ − 2φ̄ + 2σφ )3 − 2[(φ̂ − 2φ̄)+ ]3
.
48σφ2
It is easy to show that g is an increasing function of φ̂. The equilibrium price of the conglomerate, PM , is thus given by PM = 2(1 + ρ)[µ − σf g −1(γ x̂σf )/2].
Proof of Corollary 1: Without the loss of generality, we prove the case of ρ = 0, γ =
1, x = 1, σf = 1. The general case can be dealt with similarly. In the case in which σφ ≥ 6,
we have from the proof of Propositions 3 and 4, that g(φ̂) = 1 ≤ σφ /6 = g(2φ̄). Since g(·) is
a monotonously increasing function, we must have φ̂ ≤ 2φ̄. Consequently, the diversification
√
discount is 2( 3 6σφ2 −2 σφ ). The first order derivative with respect to σφ is 43 3 σ6φ − σ1φ > 0.
Similarly, in the case in which 1 < σφ < 6, we have φ̂ > 2φ̄. The diversification discount
√
is δ = φ̂ − 2φ̄ + 2σφ − 4 σφ . Substituting this back into the expression for g(φ̂), we have
√
√
F (δ) = (δ + 4 σφ )3 − 2(δ − 2σφ + 4 σφ )3 − 48σφ2 = 0.
It is easy to verify that F increases with δ. We need to prove that F decreases with σφ , then
δ must increases with σφ following the Implicit Function Theorem. First notice that the first
order derivative of F with respect to σφ is
Fσφ = 3[−δ 2 + 8(σφ −
√
φ) − 8(σφ2 − 4σφ σφ + 6σφ )].
(A4)
It is easy to verify that this derivative (A4) is less than zero when σφ < 4. When 4 ≤ σφ < 6,
√
let δ ∗ denote the lower root of Fσφ (δ), then δ ∗ = 4σφ − 4 φ − 2 2σφ2 − 8σφ . To prove that
Fσφ (δ) < 0, we need to show that δ < δ ∗ since Fσφ (δ) is a concave quadratic function of δ.
27
To prove that δ < δ ∗ , we need to show that F (δ ∗ ) > F (δ) since F is an increasing function
of δ. With some algebra, the inequality F (δ ∗ ) > F (δ) reduces to,
(4σφ − 2 2σφ2 − 8σφ )3 − 2(2σφ − 2 2σφ2 − 8σφ )3 − 48σφ2 > 0.
Define y = 2
2 − 8/σφ , then the inequality reduces to
(4 − y)3 − 2(2 − y)3 > 6(2 − y 2 /4)
which reduces to
y 3 + 33.5y 2 − 40y + 52 = y 3 + 13.5y 2 + 32 + 20(y − 1)2 > 0.
Proof of Corollary 2: Without the loss of generality, we prove the case of ρ = 0, γ =
1, x = 1, σf = 1. The more general case can be treated similarly.
First, consider σφ ≥ 6. In this case φ̂ ≤ 2φ̄. The proportion of nonparticipants is
φ∗ −φ̄+σ
[1 − 2σφ φ ]2 = (1 − 1/σφ )2 in the economy in which the two assets are traded separately.
For the economy with the conglomerate, investors with φiA + φiB < φ̂ will participate and
(φ̂−2φ̄+2σφ )2
9
3
the proportion of nonparticipants is 1 −
=
1
−
. Let x = 6 1/σφ .
2
8σ
2σ2
φ
φ
To prove the claim, we need to show that
(1 − x3 )2 < 1 − x4
which reduces to
3
9/2,
−2 + x3 + x 3 9/2 < 0.
The LHS of (A5) is an increasing function of σφ and when x =
(A5)
6
1/6, the expression holds,
therefore the corollary holds for the case in which σφ ≥ 6.
Next, we consider the case in which 1 < σφ < 6. In this case, in the economy with the
conglomerate, the proportion of nonparticipants is
(2φ̄+2σφ −φ̂)2
.
2
8σφ
φ∗ − φ̄ + σφ 2
(2φ̄ + 2σφ − φ̂)2
> [1 −
] = (1 −
8σφ2
2σφ
28
We need to show that
1/σφ )2 ,
which reduces to
√
φ̂ < φ̌ = 2φ̄ + 2σφ − 2 2σφ + 2 2σφ .
To prove φ̂ < φ̌, notice that g(φ) is an increasing function, and we need to prove that
g(φ̌) > g(φ̂) = 1.
(A6)
Substituting the expression of φ̌ into inequality (A6), we get
√
√
(4σφ − 2 2σφ + 2 2σφ )3 − 2(2σφ − 2 2σφ + 2 2σφ )3
> 1.
48σφ2
Let y =
√
(A7)
σφ , inequality (A7) reduces to
√
√
y 3 + 3( 2 + 1)y − (4 + 3 2) > 0,
(A8)
which holds since y > 1.
Proof of Theorem 1: The following information about λiA and λiB will be useful. λiA = 0,
if DiA > 0, λiA = 2σf φiA if DiA < 0 and 0 ≤ λiA ≤ 2σf φiA if DiA = 0. Left multiply both
sides of equation (22) by (µ − P + λi − φi σf ) . Then the RHS is positive and the LHS is
negative. Thus µ − P + λi − φi σf must be zero and Di = 0.
Proof of Theorem 2: Notice that 0 ≤ λiB ≤ 2σf φiB . For existence, define the aggregate
demand by the right side of equations (23) and (24) and denote it by D(P ). Using the
Maximum Theorem, it is readily shown that after the inner minimization, the objective
function in (18) is continuous and strictly concave in (DA , DB ). As a result, the demand
function D(P ) is continuous in P . Consider the following mapping
M(P ) = P + γ Ω̂(D(P ) − x) = γ Ω̂[µ − σf φ̄ + λ(P )],
where
λ(P ) =
i
λi (P ) h(φiA , φiB )dφiA dφiB .
This mapping is continuous in P . Let
S = {γ Ω̂[µ − σf φ̄ + λ], 0 ≤ λ ≤ 2σf (φ̄ + σφ )(1, 1) }.
29
Clearly, M(S) ∈ S. Applying the Brower’s fixed point theorem, there exists a fixed point in
which M(P ) = P .
To prove uniqueness, first notice that investors’ demand for portfolio A is decreasing in
PA and increasing in PB . Suppose that there exists two sets of equilibrium prices, P = P .
Without loss of generality we assume that PA > PA . Let PB∗ = PB − ω11 (PA − PA )/ω12 . We
show that at price vector P ∗ = (PA∗ , PB∗ ) = (PA , PB∗ ) the demand for asset A will be less than
∗
∗
> xA = DA . Then for some i, DiA
> DiA .
the supply. Suppose, to the contrary, that DA
∗
∗
> 0 implies λ∗iA = 0; DiA
= 0 implies
It follows that λiA ≥ λ∗iA . (This follows because DiA
∗
< 0 implies DiA < 0 which implies
DiA < 0 which implies that λiA = 2σf φiA ; and finally DiA
λiA = 2σf φiA ≥ λ∗iA .) Noting that
∗
0 > DiA − DiA
= ω11 (PA∗ − PA + λA − λ∗iA ) + ω12 (PB∗ − PB + λiB − λ∗iB ),
(A9)
either we have a contradiction in the case when ω12 = 0, or
λiB > λ∗iB ,
in the case ω12 < 0 (note that ω12 ≤ 0). In the latter case,
DiB ≤ 0,
∗
DiB
≥0
hold. It follows that
∗
0 > DiB − DiB
and hence
−ω21 (PA∗ − PA + λA − λ∗iA ) > ω22 (PB∗ − PB + λB − λ∗B ) > 0.
Combining this with equation (A9) and noting that PA∗ > PA and the two terms in the
brackets of equation (A9) are both positive, we have
ω21 ω12 > ω11 ω22 ,
which is a contradiction.
Now since the supply is nonzero, with positive measure, some investors’ demand curve
for portfolio A must be strictly increasing with PB . As a result we must have PB − PB >
−ω11 (PA − PA )/ω12 > 0. Similarly, we must have PA − PA > −ω22 (PB − PB )/ω12 > 0. These
30
two inequalities together violates the positive definiteness of Ω̂−1 . Thus the equilibrium must
be unique. Q.E.D.
Proof of Theorem 3: It is easy to show that DiA and DiB are continuous functions
of PA , PB , φiA , and φiB . Thus, λiA and λiB are also continuous functions of PA , PB , φiA ,
and φiB . We first prove that, in equilibrium, if almost all investors hold positions in both
factor portfolios, then almost all investors will hold long positions in the factor portfolios.
Suppose that the opposite is true, that is, there exists a positive measure of investors holding
short positions in at least one of the factor portfolios. Without loss of generality, suppose
that for a positive measure of investors, the demand for factor portfolio A is negative. Let
φiB = sup{φjB , DjA < 0}. Then, DiA = ω11 (µA −PA −σf φiA +λiA )+ω12 (µB −PB −σf φiB ) < 0.
Then for φjB > φiB , we must have DjA > 0 almost everywhere, and for φjB > φiB , we must
have DjA > 0 almost everywhere, and φjB < φiB , we must have DjA < 0 almost everywhere.
Thus λjA = 0 for φjB > φiB and λjA = 2σf φjA , for φjB < φiB . Thus λiA is not continuous
in φjB along the line φjB = φiB . We have a contradiction and there cannot be investors who
are holding short positions if all investors participate.
Given the condition of the supply and conjectured equilibrium price, it is easy to verify
that investors have strictly positive demand for both factors and market clears. Conversely,
if investors fully participate then the condition on the supply must be satisfied. If fully
participation equilibrium does not exist, then some investors must not participate in at least
one of the factor portfolios. With respect to partial participation, we have
Di = (γ Ω̂)−1 [µ − P + λi ] − (γ Ω̂)−1 [φi σf ] ,
(A10)
where, for k = A, B, λik is zero for positive Dik , positive for zero Dik , and 2φik σf for negative
Dik . Multiply both sides by γ Ω̂ and rearrange terms, we get
f
P −P =λ≡
i
λi h(φiA , φiB )dφiA dφiB ≥ 0,
with at least one inequality strictly holds because otherwise we get full participation. It is
then easy to see that the statement in the proposition holds.
Proof of Theorem 4: Let xA rA + xB rB denote the payoff of the merged firm. The supply
of the merged firm is thus normalized to one. Let x = (xA , xB ) , P = (PA , PB ) . Full
31
f
participation is straightforward and investors have demand equal to (x Ω̂)−1 (x ΩDi ), PM
=
x P f . Denote µM ≡ x µ. The first order condition is
f
DiM = (x Ω̂x)−1 [µM − PM
− σf x φi + λiM ].
It is easy to check that our equilibrium demand satisfies the market clearing condition.
λM
f
With limited market participation, we have P = P f + λ and PM = PM
+ λM , where
≡ i λiM h(φiA , φiB )dφiA dφiB , we show that PM < P̂M = x P .
We first prove that µA > PA . The proof for µB > PB is similar and is omitted. Since
the supply of the factor portfolios are positive, there must be some investors who hold long
positions in factor portfolio A. Let i be such an investor. Suppose that investor i also holds
a long position in factor portfolio B, we have
µ − P = γ Ω̂Di + φi > 0,
and µA − PA > 0.
Suppose that investor i holds a nonpositive position in factor portfolio B, then we must
have µB − PB > σf φjB > 0. Otherwise, we have,
(µ − P + σf φi + λi ) Di = (µ − P + σf φi + λi ) (γ Ω̂)−1 (µ − P + σf φi + λi ) ≥ 0,
while the LHS is clearly negative. Thus (µ − P + σf φi + λi ) must be zero and Di = 0, a
contradiction with the assumption that DiA > 0.
At price P̂M = x P since the risk premium is positive, it is impossible to have investors
to hold negative positions in both factor portfolios or zero in one factor portfolio and negative
in the other. Consider investor i who holds zero positions in both factor portfolios, then he
must hold zero positions in the conglomerate, thus λiM = −x (µ − P σf φi ) = x λi . If
investor i’s optimal demand are both positive, then his demand for the merged firm will be
(x Ω̂)−1 (x ΩDi ) and λiM = x λi = 0.
For other scenarios, investor i’s holding is strictly positive for at least one factor portfolio
and nonpositive for the other. Without loss of generality, suppose that investor i’s for
32
factor portfolio B is positive. Then, we must have µB − PB − σf φiB > 0, Otherwise,
multiply both sides of equation (22) by µ − P σf φi + λi , the LHS is strictly negative but the
RHS is positive. Now suppose that investor i’s demand for factor portfolio A is negative.
Then λiB = 0, λiA = 2σf φiA . If λiM = 0, then λiM < x λi . If λiM > 0, then λiM =
−x (µ − P − σf φi ) < xA (σf φiA − (µA − PA )) < 2σf φiA xA = x λi . Finally, if investor i hold
zero position in factor portfolio A, then λiA = −(µA −PA −σf φiA )+Ω12 (µB −PB −σf φiB )/Ω22 ,
and
xA λiA + xB λB = −xA (µA − PA − σf φiA ) + xA Ω12 (µB − PB − σf φiB )/Ω22
> −xA (µA − PA − σf φiA ) > −x (µ − P − σf φi ) = λiM
(A12)
Thus aggregating the demand for the conglomerate, we have
DM =
i
DiM h(φiA , φiB )dφiA dφiB = (x Ω̂x)−1 [x (µ − P − φ̄) + λM )
< (x Ω̂x)−1 [x (µ − P − φ̄ + λ)
= (x Ω̂x)−1 (x Ω̂x) = 1.
(A13)
Thus when there is limited participation, the market demand is strictly less than 1 for the
conglomerate at price x P . Since the demand is downward sloping, for the market to clear,
the conglomerate must sell at a discount.
33
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36
Figure 1: Asset Price under Limited Market Participation
1.6
1.6
1.55
Price
Price
1.5
1.4
1.3
1.45
1.4
0
0.2
0.4
0.6
0.8
Average model uncertainty
1.35
1
1.6
1.5
1.5
1.45
1.4
1.4
Price
Price
1.2
1.5
1.3
1.2
1.1
0
0.1
0.2
0.3
Supply of stock
0.4
1.35
1.3
0
1
2
3
Risk aversion
4
1.25
5
0
0.1
0.2
0.3
0.4
0.5
Model uncertainty dispersion
Asset price as a function of average model uncertainty (φ̄), stock supply (x), risk aversion
(γ), and model uncertainty dispersion (σφ ). The baseline parameter values are set at
x̄ = 0.2, µ = 1.2, σf = 0.3, ρ = 0.26795, γ = 1, φ̄ = 0.5, and σφ = 0.4.
37
Figure 2: Proportion of Nonparticipation in the Asset Markets
1
Proportion of nonparticipation
Proportion of nonparticipation
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.2
0.3
0.4
Asset payoff volatility
0.4
0.2
0
0.1
0.2
0.3
Supply of stock
0.4
0.4
Proportion of nonparticipation
Proportion of nonparticipation
0.6
0
0.5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.8
0
1
2
3
Risk aversion
4
0.3
0.2
0.1
0
5
0
0.1
0.2
0.3
0.4
0.5
Model uncertainty dispersion
Proportion of investors not participating the asset markets as a function of asset payoff
volatility (σf ), stock supply (x), risk aversion (γ), and model uncertainty dispersion (σφ ).
The baseline parameter values are set at x̄ = 0.2, µ = 1.2, σf = 0.3, ρ = 0.26795, γ = 1,
φ̄ = 0.5, and σφ = 0.4.
38
Figure 3: Decomposition of Equity Premium
0.35
Premium decomposition
Premium decomposition
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
Supply of stock
0.2
0.15
0.1
0.05
0.2
0.3
0.4
Payoff volatility
0.5
0.2
0.3
Premium decomposition
Premium decomposition
0.25
0
0.1
0.4
0.35
0.25
0.2
0.15
0.1
0.05
0
0.3
0
2
4
Risk aversion
6
0.15
0.1
0.05
0
8
0
0.1
0.2
0.3
0.4
Model uncertainty dispersion
0.5
Total equity premium (solid line), model uncertainty premium (dashed line), and risk
premium (dash-dotted line) as a function of stock supply (x), asset payoff volatility (σf ),
risk aversion (γ), and model uncertainty dispersion (σφ ). The baseline parameter values are
set at x̄ = 0.2, µ = 1.2, σf = 0.3, ρ = 0.26795, γ = 1, φ̄ = 0.5, and σφ = 0.4.
39
Figure 4: Diversification Discount
0.35
0.3
Price discount
0.25
0.2
0.15
0.1
0.05
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Model uncertainty dispersion
0.35
0.4
0.45
0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
Model uncertainty dispersion
0.35
0.4
0.45
0.5
9
8
Percentage price discount
7
6
5
4
3
2
1
0
Diversification discount in level (top panel) and in percentage (bottom panel) as a function
of model uncertainty dispersion (σφ ). The baseline parameter values are set at x̄ = 0.2,
µ = 1.2, σf = 0.3, ρ = 0.26795, γ = 1, φ̄ = 0.5, and σφ = 0.4.
40
Figure 5: Nonparticipation with and without Merger
0.7
0.6
Nonparticipation rate
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Model uncertainty dispersion
0.35
0.4
0.45
0.5
Nonparticipation without (solid) and with (dashed) merger as a function of model
uncertainty dispersion (σφ ). The baseline parameter values are set at x̄ = 0.2, µ = 1.2,
σf = 0.3, ρ = 0.26795, γ = 1, φ̄ = 0.5, and σφ = 0.4.
41