Model Uncertainty, Limited Market Participation
and Asset Prices
H. Henry Cao, Tan Wang and Harold H. Zhang∗
This Version: July 14, 2003
∗
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, (919) 962-5913,
zhangha@kenan-flagler.unc.edu, (919) 843-8340; and Tan Wang is with the Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, British Columbia, Canada, V6T 1Z2,
[email protected], (604) 822-9414. The authors are grateful to seminar participants at the economics department of the University of North Carolina at Chapel Hill, the finance group of Cheung-Kong
Graduate School of Business, Cornell University, Duke University, Hong Kong University of Science and
Technology, National University of Singapore, New York University, Rice University, University of Arizona,
University of California at Irvine, University of California at Los Angeles, University of California at Riverside, University of Georgia at Athens, University of Minnesota, University of North Carolina at Chapel Hill,
University of North Carolina at Charlotte, and University of Texas at Austin, for helpful comments. Tan
Wang thanks the Social Sciences and Humanities Research Council of Canada for financial support.
Model Uncertainty, Limited Market Participation
and Asset Prices
Abstract
We demonstrate that limited participation can arise endogenously in the presence
of model uncertainty and heterogeneous uncertainty averse investors. Our model generates novel predictions on how limited participation relates to equity premium and
diversification discount. When the dispersion in investors’ model uncertainty is small,
full participation prevails in equilibrium. In this case, equity premium is unrelated to
uncertainty dispersion and a conglomerate trades at a price equal to the sum of its
single segment counterparts. When uncertainty dispersion is large, however, investors
with relatively high uncertainty optimally choose to stay sidelined in equilibrium. In
this case, equity premium can decrease with uncertainty dispersion. Our results suggest that the understanding in the existing literature that limited participation leads
to higher equity premium is not robust. Moreover, when limited participation occurs,
a conglomerate trades at a discount relative to its single segment counterparts. The
discount increases in uncertainty dispersion and is positively related to the proportion
of investors not participating in the markets.
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 how limited participation may arise in an
equilibrium and its implications for equilibrium asset prices. Specifically, we demonstrate
that (1) limited participation may arise due to investors’ uncertainty about the distribution
of asset payoffs; (2) equity premium could be lower in the economy in which there is limited
participation than in the economy in which there is full participation; and (3) when there
is limited participation, a merged firm is traded at a value lower than the sum of values
of its component firms traded separately, i.e., there can be a diversification (conglomerate)
discount.
Several models have been proposed to explain why limited market participation may
exist. Allen and Gale (1994), Williamson (1994), Vissing-Jorgensen (1999), and Yaron and
Zhang (2000) focus on how entry costs and/or liquidity needs create limited market participation. However, it is difficult for these models to explain why families with substantial liquid
assets (over $100,000) may not participate in stock markets. Haliassos and Bertaut (1995)
also investigate possible explanations for limited market participation and demonstrate that
risk aversion, heterogeneous beliefs, habit persistence, time-nonseparability, borrowing constraints, differential borrowing and lending rates, and minimum-investment requirements are
unable to account for the phenomenon. They argue informally that fixed cost and departure
from expected utility maximization are more promising explanations.1
1
In partial equilibrium models, Dow and Werlang (1992) provides an explanation why some investors
may not participate in the stock market based on uncertainty aversion while Ang, Bekaert and Liu (2002)
1
Our study offers an explanation for limited market participation based on investors’
heterogeneity in model uncertainty. Compared to models based on fixed costs, our model
can also explain why families with substantial liquid assets do not participate in the stock
markets. The theoretical prediction of our model on market participation is consistent with
the empirical evidence documented in Blume and Zeldes (1994) and Haliassos and Bertaut
(1995).
The implications of limited participation on equity premium has been examined in several studies. For instance, Mankiw and Zeldes (1991), Vissing-Jorgensen (1999), and Brav,
Constantinides, and Geczy (2002) empirically demonstrate that if the sub-sample 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), thus helps to resolve the equity premium
puzzle. Basak and Cuoco (1998) show theoretically that equity premium increases as more
investors are excluded from the risky asset markets.2 The limited market participation in
these studies, however, is exogenously specified. Moreover, these studies are all based on
standard expected utility maximization models which abstract from model uncertainty. In
our framework, investors face both risk and model uncertainty. The equilibrium equity premium can be decomposed into the risk premium and the uncertainty premium. When limited
market participation arises endogenously, the reduced market participation could drive uncertainty premium down more than it drives risk premium up, leading to a lower total equity
premium. Our results suggest that the conventional view that limited participation leads to
higher equity premium is not robust when market participation is endogenous.
In the corporate finance literature, the diversification discount phenomenon has been
documented in a number of studies. For example, Lang and Stulz (1994), Berger and Ofek
(1995), Rajan, Servaes, and Zingales (2000) show that conglomerates exhibit on average
negative excess value of 10-15%.3 As the flip side of the conglomerate discount, Hite and
argue that disappointment aversion can produce similar results.
2
Shapiro (2002) extends Basak and Cuoco’s model to multiple stocks.
3
Currently, there is a debate as to how significant diversification discount is. For example, a recent study
by Bevelander (2002) finds that most diversified firms have Tobin’s qs comparable to those of focused firms
when firm age is explicitly accounted for. Villanoga (2002) documents that diversified firms actually trade
2
Owers (1983), Miles and Rosenfeld (1983) and Schipper and Smith (1983) document a 2-3%
increase in shareholder value around the corporate spin-offs. Rajan, Servaes, and Zingales
(2000) and Scharfstein and Stein (2000) argue that diversification discount can be attributed
to inefficient internal capital markets and/or inefficient resource allocation.4 We show that
diversification discount may arise due to investor heterogeneity in uncertainty about firms’
asset returns and is closely associated with limited participation. Our study establishes
relations among the diversification discount, investor heterogeneity, and limited participation
from an asset pricing perspective.5
We perform our analysis in a tractable single period mean-variance framework with heterogeneous agents. The added feature of our study is that, in addition to the risk associated
with asset payoffs, investors are uncertain about the distribution of stock payoffs. That is,
there is model uncertainty. In his seminal work, Knight (1921) argues that risk, a situation where the probability distribution is known, is fundamentally different from uncertainty
where the distribution is unknown. Such difference is now well recognized in both the academic literature (Ellsberg (1961)) and the investment industry. As one wall street journalist
put it (Lowenstein (2000)), “there is a key difference between a share of IBM ... and a
pair of dice. With dice, there is risk—you could, after all, roll snake eyes—but there is no
uncertainty, because you know (for certain) the chances of getting a 7 and every other result.
Investing confronts us with both risk and uncertainty. There is a risk that the price of a
share of IBM will fall, and there is uncertainty about how likely it is to do so.”
Our study follows the Knightian approach to model uncertainty which differs from the
more common Bayesian approach. In the Bayesian approach, when facing uncertainty about
model specifications, the investor uses a prior distribution over a set of models.6 Thus
the investor’s preference is represented by a Savage (subjective) expected utility. In the
at a significant premium using the Business Information Tracking Series (BITS) data which covers the whole
U.S. economy at the establishment level.
4
Recent studies by Graham, Lemmon and Wolf (2002) and Maksimovic and Phillips (2002) did not find
empirical evidence that would suggest inefficient resource allocation across divisions of a conglomerate.
5
There are alternative theories about the diversification discount and spin-off premium. For example,
Miller (1977) argues informally that 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. Gomes and
Livdan (2002) develop a dynamic model of firm diversification focusing on the role of diminishing marginal
productivity of technology and synergies of different activities in reducing fixed cost of production.
6
Wang (2002) shows that an investor’s portfolio decision is highly sensitive to the choice of prior.
3
Knightian approach to uncertainty,7 while investors may still use von-Neumann-Morgenstern
expected utility when facing risk, they no longer use a single prior when facing uncertainty.
In a seminal paper, Gilboa and and Schmeidler (1989) operationalize Knightian uncertainty
using a multi-prior expected utility framework. In this framework, an uncertainty averse
investor evaluates an investment strategy according to the expected utility with the worst
case probability distribution in the set of priors distributions. Thus, an uncertainty averse
investor will be more conservative when he faces more uncertainty.8
In our model, investors are uncertain about the mean of risky asset payoffs and are heterogeneous in their levels of uncertainty.9 The crucial aspect of our framework, pertaining
to the Knightian approach in modeling investors’ imperfect knowledge of the payoff distribution, is that for each uncertainty averse investor there is a region of prices over which the
investor neither buys nor sells short the stocks. In the single stock case, the nonparticipation
region of prices lies between the lowest and the highest expected payoffs of the investor’s
admissible payoff distribution set. For any price in the nonparticipation region, the investor
will not take a long position in the stock, as he uses the lowest expected payoff to evaluate
this strategy. Similarly the investor will not take a short position as he uses the highest
expected payoff in this case. Investors who are more uncertain about the mean payoff of
the stock have larger nonparticipation regions than investors with less uncertainty about the
mean payoff. As a result, when uncertainty dispersion is sufficiently high, in equilibrium,
investors with relatively high uncertainty choose not to participate in the stock market,
giving rise to limited participation.10 Further, more investors choose to stay sidelined as
uncertainty dispersion increases.
7
See 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 (2003), Routledge
and Zin (2001), Liu, Pan and Wang (2002), among others.
8
For alternative specifications of Knightian uncertainty, see Schmeidler (1989) and Bewley (2002). In
Schmeidler (1989), an agent evaluates a choice according to its expected utility using a capacity (Choquet
expected utility with non-additive probability). Bewley (2002) 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.
9
This modeling strategy of uncertainty has been used in a number of studies including Epstein and Miao
(2003), Uppal and Wang (2001), Kogan and Wang (2002), Liu, Pan and Wang (2002) among others. Kogan
and Wang (2002) is the closest to ours in terms of modeling framework but we consider an equilibrium model
with heterogeneous agents.
10
Models based on Bayesian theory such as Williams (1977) cannot generate limited participation, unless
there is short-sale constraint or investors who are sidelined believe that the equity premia are zero.
4
The intuition for the implication of limited market participation for equity premium
can be explained as follows. In our model, equity premium is decomposed into the risk
premium and the uncertainty premium. The risk premium corresponds to the equity premium obtained in standard expected utility models. The uncertainty premium arises due
to the uncertainty that investors have about the distribution of the stock payoff. When the
uncertainty dispersion is high, only investors with low uncertainty participate in the stock
markets.11 As a result, there are two forces affecting equity premium when limited market
participation occurs. On the one hand, since fewer investors hold the stocks, the equilibrium risk premium has to be higher to induce those investors to hold the stocks so that the
markets clear. On the other hand, because investors participating in the stock markets have
lower uncertainty, a lower uncertainty premium is sufficient to induce these investors to hold
the stocks. Consequently, the net effect of an increase in uncertainty dispersion on equity
premium depends on the relative dominance of these two opposing forces. We find that the
decrease in the uncertainty premium outweighs the increase in the risk premium, leading
to a lower equity premium than that in the economy with full market participation. Furthermore, equity premium decreases (increases) as model uncertainty becomes more (less)
dispersed and more (less) investors stay sidelined.
The intuition behind the relation between limited participation and diversification discount is also straightforward. When there is limited participation at firm level, the premerger price of an individual firm is determined by investors with less uncertainty for that
firm. They therefore demand a lower uncertainty premium and are willing to pay a higher
price for the firm’s stock. 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
the prices of the two firms traded separately. Thus, when there is limited participation, a
merger results in a discount in the price of the combined firm.12
The rest of the paper is organized as follows. Section 2 presents the model. Section 3
11
In standard models, risk aversion alone cannot explain why investors have zero rather than small holdings
in stocks.
12
Since we want to focus on the link between limited participation and diversification discount, we have
abstracted from the production decisions of the firms. Gomes and Livdan (2003) show how a diversification
discount can be generated in a model with endogenous production and industrial equilibrium. They do not
consider how limited participation is related to diversification discount.
5
describes investors’ portfolio choices and their participation decisions. Section 4 analyzes
how the limited market participation relates to equity premium, while Section 5 extends the
model to allow for two stocks and investigates the relation between limited participation
and the diversification discount. Section 6 discusses the robustness of our results in a more
general setting with richer factor structure for asset payoffs. Section 7 concludes. Proofs
and some technical details are collected in the appendix.
2
The Model
To highlight the intuition, we first consider a market with one stock and one bond. The case
of multiple stocks is considered in Section 5.
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 one stock and one bond in the economy. Investors are endowed with shares of the stock.
The per capita supply of the stock is denoted x > 0. The risk-free rate is set at zero. The
stock can be viewed as the market portfolio in a single factor economy in which the market
portfolio is the only factor. The payoff of the stock, denoted r, is normally distributed with
mean µ and variance σ 2 . We assume that investors have a precise estimate of the variance
σ 2 , but do not know exactly the mean of the stock payoff r. This is motivated by both the
analytical tractability and the empirical evidence on the strong predictability of the volatility
of stock returns, for instance using ARCH and GARCH models, but the unpredictability of
the level of stock returns. The imperfect knowledge of the stock payoff distribution gives
rise to model uncertainty.
2.2
The Preferences
Each agent in the economy has a CARA utility function u(W ), where W is the end of period
wealth. Due to the lack of perfect knowledge of the probability law of stock payoff, the
6
agent’s preference is represented by a multi-prior expected utility (Gilboa and Schmeidler
(1989))
min EQ [u(W )] ,
Q∈P
(1)
where EQ denotes the expectation under the probability measure Q and P is a set of probability measures. The general nature and the axiomatic foundation of these preferences have
been examined in detail in Gilboa and Schmeidler (1989). The basic intuition behind this
preference is that in the real world, people do not know precisely the probability distribution
of stock payoff. They may have a reference model of the probability distribution, such as
a normal distribution with certain mean and variance, which could be the result of some
econometric analysis. However, they are not completely sure that the reference model is
the right model. For example, there may be uncertainty about the estimate of the mean
stock payoff. As a result, instead of restricting themselves to using distribution P , they
are willing to consider a set P of alternative probability distributions. 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 in the set P.
The specification of the set P follows Kogan and Wang (2002) closely. In the interest of
focusing on the issues being studied, we provide a brief description of P and its intuition.
More details can be found in Kogan and Wang (2002). Let µ be the point estimate of the
mean stock payoff. One way to quantify the uncertainty is by using the confidence region. For
example, if P is the reference probability distribution., the distribution resulted from econometric analysis, and Q is an alternative measure, then the inequality E P [− ln(dQ/dP )] < η
defines a confidence region according to the log likelihood ratio where η depends on the
investor’s uncertainty on the mean stock payoff.13 By choosing appropriate η, the region can
yield the desired level of confidence, for example, 95%. In other words, if P is defined to be
the set of all Q that satisfies the inequality, then with 95% confidence, we can reject any Q
outside P being the right model or the probability that the right model lies outside P is no
more than 5%.14
13
As Kogan and Wang (2002) show, the investor’s aversion to uncertainty can also be embedded in η. For
ease of exposition, we assume that all investors in our economy have the same level of uncertainty aversion.
Of course, allowing for heterogeneous uncertainty aversion will strengthen our results.
14
In the case of multiple risky assets, investors may have different levels of uncertainty on the mean asset
payoff and aversion to the uncertainty.
7
Kogan and Wang (2002) show that for normal distributions with a common known
variance, the confidence region can always be described by a set of quadratic inequalities.
For the specific case of this section, it takes the form of (µ+v) where v denotes the adjustment
to the estimated mean stock payoff and satisfies:
v 2 σ −2 ≤ φ2 ,
(2)
where φ is a parameter that captures the investor’s uncertainty about the mean stock payoff.
We therefore define the set P as the collection of all normal distributions with mean µ + v
and variance σ 2 where v satisfies (2). All investors are assumed to have the same point
estimate µ.15 Note that the higher the φ, the wider the range for the expectation of r. Thus
we call φ the investor’s level of uncertainty on the mean stock payoff.
In our economy, investors have the same risk aversion, but are heterogeneous in their
levels of uncertainty.16 Such heterogeneity can arise for a variety of reasons. For example,
the data collected by the research department of investment banks and asset management
firms are private and of different quality. Different firms are likely to have different data
sets about the same stock, which may lead to differences in estimated payoff distributions.
Different firms also use their own proprietary models to analyze the data. This may also
lead to different estimated payoff distributions.17 We abstract from the details of how such
heterogeneity arises and simply assume that the differences are captured by different φ.
Furthermore, we assume that φ is uniformly distributed among investors on the interval18
S1 = [φ̄ − δ, φ̄ + δ],
where φ̄ ≥ δ and δ measures the dispersion of the uncertainty among investors. When
φ̄ = δ = 0, there is no model uncertainty and our model collapses to the standard expected
utility model.
It is important to note that the preferences represented in (1) exhibit aversion to uncertainty whereas preferences represented by Savage’s expected utility are uncertainty neutral.
15
Our results regarding limited participation, equity premium, and diversification discount hold without
assuming the same point estimate for all investors.
16
In an early version, we allowed for heterogeneity in investors’ risk aversion. Because our results are not
affected, we assume the same risk aversion for all investors for the ease of exposition.
17
Differences in perceived familarity about the stock market may contribute to different degrees of uncertainty.
18
In Section 6, we consider more general distribution for uncertainty among investors.
8
The simplest example that highlights the difference is the Ellsberg experiment (Ellsberg
(1961)). It can be readily checked that no matter what prior distribution is used to assess
the probability of drawing a red ball from the urn with unknown proportion of each color
(the second urn), to be consistent with the behavior exhibited in the Savage expected utility,
the predictive probabilities of drawing a red or a white must be equal. This implies that a
decision maker should be indifferent between betting red on the first urn and betting red
on the second urn. In other words, the fact that the probability of drawing a red from the
second urn is unknown does not make any difference in his decision making. Such neutrality
to model uncertainty, however, does not hold for preferences represented by (1). In fact,
decision makers with such preferences will always prefer betting red on the first urn to betting red on the second urn. This difference between the multi-prior expected utility and the
standard expected utility is critical in understanding why our findings differ from the results
of existing studies.
3
Equilibrium Market Participation
In this section, we first solve for investors’ optimal portfolio choice and then find the closedform solution for the equilibrium. We show that when the dispersion in investor’s uncertainty
is high, there is limited market participation in equilibrium.
3.1
Portfolio Choices
Let P denote the price of the stock, the investor’s wealth constraint is
W1i = W0i + Di (r − P ),
where W0i and W1i are investor i’s beginning of period and end of period wealth, respectively,
Di is investor i’s demand for the stock. Under the assumption of normality and the CARA
utility, investor i’s utility maximization problem becomes:
max min Di (µ − v − P ) −
Di
v
9
γσ 2 2
D ,
2 i
where γ is the investor’s risk aversion coefficient. The first term of the objective function
represents the expected excess payoff from investing in the stock with adjustment to the
mean due to uncertainty, and the last term represents adjustment due to risk. Solving the
inner minimization, the above problem reduces to
γσ 2 2
D ,
max Di [µ − sgn(Di )φi σ − P ] −
Di
2 i
(3)
where sgn(·) is an indicator function which takes the sign of its argument. The expression,
µ − sgn(Di )φi σ, represents the uncertainty-adjusted expected payoff under the worst case
scenario for an investment strategy of Di shares in the stock. The intuition behind (3) is
the following. If investor i takes a long position (Di > 0 and sgn(Di ) = 1), the worst case
scenario for him is that the mean excess payoff is given by µ − φiσ − P . Similarly, if he takes
a short position (Di < 0 and sgn(Di ) = −1), the worst case scenario is that the mean excess
payoff is µ + φi σ − P .
The investor’s optimal holding in the stock is given by


1
(µ
γσ2
− φi σ − P ) if µ − P > φi σ,
Di =
0
if − φi σ ≤ µ − P ≤ φiσ,
 1
(µ + φi σ − P ) if µ − P < −φi σ.
γσ2
(4)
It can be seen from (4) that, due to his uncertainty about the mean of the stock payoff,
investor i buys (sells short) the stock only when the equity premium is strictly positive
(negative) in the worst case scenario, v = φi σ (v = −φi σ). When the price P falls in the
region, [µ − φi σ, µ + φi σ], the investor optimally chooses not to participate in the market.
The nonparticipation region for investor i is the interval between the investor’s lowest and
the highest expected payoffs. For any price in the region, investor i will not take a long
position because his uncertainty-adjusted expected payoff under the worse case scenario for
a long position is too low. He will not take a short position either as his uncertainty-adjusted
expected payoff under the worst case scenario for a short position is too high. This is in
sharp contrast to the standard result based on expected utility models that as long as the
equity premium is not zero (µ − P = 0), the investor will hold a nonzero position in the
stock.
10
3.2
Full Participation
In equilibrium, the market clears. The supply of the stock is equal to its aggregate demand.
As will be seen later, in equilibrium, the stock always sells at a discount, i.e., µ − P > 0.
Given this, the expression of the demand function (4) implies that no investor will hold short
positions in the stock. In the full participation equilibrium, all investors participate in the
stock market. The market clearing condition for the stock can thus be written as
x=
φi ∈S1
(µ − σφi − P ) 1
1
(µ − σ φ̄ − P ).
dφi =
2
γσ
2δ
γσ 2
Solving for P , we arrive at the following equilibrium price for the stock,
P = µ − γσ 2 x − σ φ̄.
(5)
As long as there exists model uncertainty (φ̄ > 0), the equilibrium stock price will be less
than the expected stock payoff, i.e., µ − P > 0.
We can rewrite Equation (5) to express the equity premium as follows
µ − P = γσ 2 x + σ φ̄.
(6)
The first term on the right hand side of Equation (6) is standard. It represents the risk
premium, which is proportional to investors’ risk aversion coefficient, the variance of the stock
payoff, and per capita stock supply. The second term is new. It represents the uncertainty
premium and is proportional to the average level of uncertainty (φ̄) of all investors about
the mean stock payoff.
The full participation equilibrium prevails when all investors participate in the stock
market. This requires all investors, including investors with the highest uncertainty, to hold
long positions. Equation (4) thus implies
µ − (φ̄ + δ)σ − P > 0.
Substituting (5) into the expression, we have that in equilibrium all investors participate in
the market if
γσ 2 x > δσ.
11
(7)
This full participation condition indicates that whether all investors participate in the stock
market depends on the dispersion of investors’ model uncertainty. When δ is small, investors
are reasonably homogeneous and all investors will participate. It also depends on the Sharpe
ratio, (γσ 2 x)/σ. Given the model uncertainty, the higher the Sharpe ratio, the more likely
investors with the highest uncertainty will participate in the stock market.
It is interesting to note that under full market participation, investors’ model uncertainty
dispersion has no effect on price. What matters is the average uncertainty. Investors who are
more uncertain about the mean stock payoff will hold less of the stock, while investors who
are less uncertain will hold more. In aggregate, the market price behaves as if all investors
have the average uncertainty.
3.3
Limited Participation
When the uncertainty dispersion among investors is sufficiently high such that condition (7)
is not satisfied, investors with high uncertainty optimally choose not to participate in the
stock market. Let φ∗ denote the lowest level of uncertainty at which investors do not hold
the stock. Then
µ − σφ∗ − P = 0.
(8)
For an investor with uncertainty higher than φ∗ , he chooses not to participate in the stock
market. Thus, the market clearing condition is given by
x=
φi <φ∗ ,φi ∈S1
σ(φ∗ + φ̄ − δ)
Di
1 (φ∗ − φ̄ + δ)
dφi =
µ−
−P .
2δ
γσ 2
2δ
2
Solving for φ∗ yields
φ∗ = φ̄ − δ + 2 γδσx.
(9)
(10)
Using Equation (8), we arrive at the following equilibrium price for the stock:
P = µ − σ(φ̄ − δ) − 2σ
γδσx.
(11)
Once again, in the presence of model uncertainty, the stock is traded at a discount relative
to its expected payoff.19
19
Note that φ̄ − δ ≥ 0 by definition.
12
To gain intuition of this pricing formula, we rewrite Equation (9) as
µ−P =
γσ 2 x
+ φp σ,
α
where
φ∗ − (φ̄ − δ)
α=
=
2δ
γσx
,
δ
(12)
(13)
is the proportion of investors holding long positions in the stock, and
1
φp = (φ∗ + φ̄ − δ) = φ̄ − δ + γδσx,
2
(14)
is the conditional average level of uncertainty about the mean stock payoff among market
participants. Because market participants have lower uncertainty than nonparticipants, the
conditional average uncertainty (φp ) is lower than the unconditional average uncertainty (φ̄).
Equation (12) suggests that the equity premium can be decomposed into two components:
the risk premium and 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 as the uncertainty dispersion (δ) increases. Consequently, lower uncertainty
premium is associated with lower participation rate.
Figure 1 provides some simple comparative static results with respect to average model
uncertainty (φ̄), stock supply (x), risk aversion (γ), and model uncertainty dispersion (δ).
The stock price decreases in average model uncertainty and the supply. Given the stock
payoff, higher model uncertainty and supply make the stock less attractive to investors,
which leads to a lower price. When the supply is very low (x is very small) such that there is
limited participation (condition (7) does not satisfy), the stock price becomes very sensitive
to the change in supply. This is reflected in steeper pricing function as the supply decreases.
When the supply is very large, there is full participation and the price becomes linear in
supply. The stock price also decreases in investors’ risk aversion. Intuitively, investors with
low risk aversion are more likely to buy a stock for the same future payoff than investors with
high risk aversion. As investors’ risk aversion increases, the demand for the stock decreases.
This leads to lower price for the stock.
13
As uncertainty dispersion increases, investors who are more uncertain about the mean
payoff of the stock gradually withdraw from the market, leaving those who are less uncertain
in the market. Because these remaining investors have lower uncertainty and are willing
to pay higher price for the stock, the equilibrium stock price increases. Note that when
uncertainty dispersion is low, the stock price does not vary with uncertainty dispersion.
This is because at low uncertainty dispersion there is full market participation and the
equilibrium price is determined by the average uncertainty and not affected by the dispersion
of uncertainty.
In Figure 2 we plot the percentage of nonparticipation in the stock market, (1 − α), as
a function of stock payoff volatility (σ), stock supply (x), risk aversion (γ), and uncertainty
dispersion (δ). The percentage of investors not participating in the stock market decreases
as the payoff volatility, stock supply, and investors’ risk aversion increase. This is caused by
lower price associated with higher payoff volatility, stock supply, and risk aversion. On the
other hand, because the price is higher at higher dispersion of model uncertainty, investors
are increasingly staying sidelined as model uncertainty dispersion increases. This finding is
in contrast to the result in the standard expected utility model in which investors always
take some positions in the stock when there is a positive equity premium.
4
Market Participation and Equity Premium
A widely investigated issue in the literature on limited market participation is how equity
premium relates to limited market participation. Several studies (Mankiw and Zeldes (1991),
Basak and Cuoco (1998), Vissing-Jorgensen (1999), and Brav, Constantinides, and Geczy
(2002)) find that equity premium increases as fewer investors participate in stock markets.
These studies, however, assume that some investors are exogenously excluded from participating in certain markets. In this section, 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 so that limited market participation arises endogenously
in our economy.
The following is the main result of this section.
14
Proposition 1 If γσx < δ, then there is limited market participation in equilibrium. Further, in the presence of limited market participation,
(a) investors’ participation rate α decreases with uncertainty dispersion, i.e., ∂α/∂δ < 0;
(b) the average model uncertainty for investors holding positive positions in the stock decreases with uncertainty dispersion, i.e., ∂φp /∂δ < 0; and
(c) the equity premium decreases with uncertainty dispersion, i.e., ∂(µ − P )/∂δ < 0.
We have shown earlier that with full market participation, equity premium does not
depend upon model uncertainty dispersion among investors. With limited market participation, however, equity premium decreases with model uncertainty dispersion. The intuition is
as follows. As uncertainty dispersion increases, investors with relatively high uncertainty will
not participate in the stock market. In this case, an investor taking a long position uses the
lowest expected payoff in his payoff distribution set to evaluate his investment strategy and
an investor taking a short position uses the highest expected payoff to evaluate his investment strategy while a sidelined investor uses the expected payoff that lies between the two
extremes. When there is full market participation, every investor uses the lowest expected
payoff in his payoff distribution set to evaluate his investment strategy. Given the same
average uncertainty, the aggregate demand function for the stock is thus higher when there
is limited participation than in the economy with full participation. As a result, the equilibrium price is higher and the equity premium is lower under limited participation than under
the full participation. This can be readily seen from Equation (12). As uncertainty dispersion 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 stock payoff and are willing to accept lower uncertainty premium. Moreover, in our model, when there is limited market participation, the uncertainty premium
effect dominates the risk premium effect, leading to a lower equity premium as shown in
Proposition 1(c). Since more investors choose to stay sidelined when the uncertainty dispersion is higher, our results imply that lower (higher) equity premium is associated with less
(more) stock market participation.
15
There is some empirical support for the predicted positive relation between stock market
participation and stock returns. In Figure 3, we plot the stock returns against the stock
market participation for seven countries including France, Germany, Italy, the Netherlands,
Sweden, the United Kingdom, and the United States. The stock returns are total annual
stock returns in US dollars for the period between 1986 and 1997. Two measures of the
stock market participation are used: direct participation which considers traded and nontraded stocks held directly and total participation which also includes mutual funds and
managed investment accounts.20 The stock market participation measures are based on
1998 microeconomic surveys in respective countries except for Sweden which is based on a
1999 survey. Overall, the figure suggests that stock returns are positively related to both
measures of stock market participation.
Our theoretical prediction also is consistent with some recent empirical studies on the asset pricing effect of the breadth of stock ownership. Specifically, if narrower stock ownership
is used as a proxy for lower participation at the firm level, our prediction that lower participation is associated with lower equity premium is consistent with the finding in Chen, Hong
and Stein (2002) that a narrower breadth of stock ownership is related to lower expected
stock returns.21
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 stock
payoff, the investors’ risk aversion, average model uncertainty, and uncertainty dispersion.
Proposition 2 Both the uncertainty premium and the risk premium increase with the variance of the stock payoff (σ) and the risk aversion coefficient (γ). The risk premium is
unrelated to the average model uncertainty (φ̄) but increases with the uncertainty dispersion
(δ). However, the uncertainty premium increases with the average model uncertainty (φ̄) but
decreases with the dispersion uncertainty (δ).
To gauge the relative importance of the two components of the equity premium, we
conduct some numerical analysis. Figure 4 shows the equity premium (solid line) decomposed
20
The data are from Tables 4 and 11 of Guiso, Haliassos and Jappelli (2002). For Germany, only direct
participation measure is available.
21
Chen, Hong and Stein (2002) interpret their empirical results using Miller’s model which is based on
heterogeneous beliefs and short-sale constraint.
16
into the uncertainty premium (dashed line) and the risk premium (dash-dotted line) as a
function of the stock supply (x), the payoff volatility (σ), the investor’s risk aversion (γ),
and the uncertainty dispersion among investors (δ). Both the uncertainty premium and the
risk premium as well as the total equity premium are increasing in the stock supply and
the volatility of stock payoff. This reflects the fact that investors demand higher returns
to hold more stock 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 stock.
Furthermore, a higher risk aversion implies a higher conditional average uncertainty among
the market participants (φp ), hence higher uncertainty premium. Notice that the uncertainty
premium initially stays flat and then decreases as the model uncertainty becomes more
dispersed among investors. This happens because there is full market participation at low
levels of uncertainty dispersion. The uncertainty premium at full market participation only
depends upon the average level of model uncertainty and is not affected by the uncertainty
dispersion. As the uncertainty dispersion increases, investors with high uncertainty stay
sidelined, which creates limited market participation. When limited market participation
occurs, the uncertainty premium decreases with the uncertainty dispersion. In fact, the
decrease in uncertainty premium outweighs the increase in the risk premium so that the
total equity premium also decreases.
5
Multiple Stocks and the Diversification Discount
In this section, we examine how, in the absence of any synergy between two firms, the
market value of the merged firm (the conglomerate) depends upon investors’ uncertainty
dispersion and how it relates to limited participation at firm level. Our study differs and
also complements studies based on production and synergy such as Gomes and Livdan (2002)
by focusing on the demand for risky assets. For our purpose, we extend the model in Section
2 to allow for two stocks. The payoffs on the two stocks follow a two factor model
r1 = fA + ρfB ,
(15)
r2 = ρfA + fB ,
(16)
17
where r1 and r2 are payoffs on stock 1 and 2, respectively, fA and fB are two (random)
systematic factors, and ρ represents the factor loading. We assume |ρ| < 1. The total
supplies of the two stocks are given by x1 and x2 , with x1 = x2 = x > 0. All variables
are jointly normally distributed. Investors know that the factors, fA and fB , follow a joint
normal distribution with a non-degenerate variance-covariance matrix ΩF but do not have
perfect knowledge of the distribution of the random factors. As in Section 2, we assume
that investors have a precise estimate of the variance-covariance matrix (ΩF ). Without loss
of generality, we assume that ΩF is diagonal with common element σ 2 . However, investors
do not know exactly the mean of fA or fB . Let µA = µB = µ be the reference level for the
mean of fA and fB . 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 :
vA2 σ −2 ≤ φ2A ,
(17)
vB2 σ −2 ≤ φ2B ,
(18)
and
where φA and φB are parameter that captures the investor’s uncertainty about the mean of
factor A and B, respectively. In other words, investors believe the mean of factor fA and fB
fall into the set
{(µA + vA , µB + vB ) : vA and vB satisfy (17) and (18), respectively}.
We assume that φ = (φA , φB ) are uniformly distributed on the square22
S2 = [φ̄ − δ, φ̄ + δ] × [φ̄ − δ, φ̄ + δ],
where φ̄ ≥ δ.
When there is a merger, firms combine their operations and cash flows to form a conglomerate denoted M. The cash flows are unaffected by the merger so that the payoff from
firm M is rM = r1 + r2 = (1 + ρ)(fA + fB ).23 After the merger, investors can no longer trade
stocks 1 and 2 and can only hold asset M.
22
More general distributions are considered in Section 6. Implicitly, we assume that investors may have
different degrees of uncertainty about different factors. For example, an American investor may have less
uncertainty about the US stock returns than the Japanese stock returns while a Japanese investor may
have less uncertainty about the Japanese stock returns than the US stock returns. Perceived familiarity of
different stocks may affect the degree of uncertainty about these stocks (Huberman (2001)).
23
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).
18
For ease of exposition, we introduce factor portfolios tracking the two factors A and B.
Let DiA and DiB be investor i’s holdings of the factor portfolio A and B, respectively. Let
PA and PB be the prices of factor portfolio A and B, respectively. Let Pj denote the price
of the stock j, Dij denote investor i’s demand for stock j, for j = 1, 2. Let DiM be investor
i’s demand for the merged firm and PM be the market price of the merged firm. Given the
payoff structure for stocks, the payoffs for the factor portfolios A and B are given by:
fA ≡
r1 − ρr2
,
1 − ρ2
fB ≡
r2 − ρr1
,
1 − ρ2
and the demand for the factor portfolios, DiA and DiB , can be written as:
DiA = Di1 + ρDi2 ,
DiB = ρDi1 + Di2 .
Similarly, we can rewrite the demand for the stocks in terms of the demands for the factor
portfolios:
Di1 =
DiA − ρDiB
,
1 − ρ2
Di2 =
DiB − ρDiA
.
1 − ρ2
The supply of each factor portfolio is x̂ ≡ (1 + ρ)x. Since the two factors are orthogonal
to each other and investors have CARA utility function, the prices of the factors (PA and
PB ) can be determined similarly as in the case with a single stock in Section 3. The prices
of the stocks when traded separately can be obtained using the relation P1 = PA + ρPB and
P2 = ρPA + PB .
5.1
Full Participation
We first consider the case of full participation. The following proposition summarizes the
main results under full participation.
19
Proposition 3 When (1 + ρ)xγσ ≥ δ, there is full participation in the economy with or
without merger. Moreover, the prices of the two stocks in the economy without merger are:
P1 = P2 = (1 + ρ)[µ − σ φ̄ − γ(1 + ρ)xσ 2 ].
(19)
and the price for the conglomerate in the economy after the merger is given by
PM = P1 + P2 = 2(1 + ρ)[µ − σ φ̄ − γ(1 + ρ)xσ 2 ].
(20)
When (1 + ρ)xγσ ≥ δ, there is full participation in both the economy with separate
trading of the two stocks and the economy with only the merged firm being traded. Equation
(20) indicates that the price of the merged firm is simply the sum of the market values of
the two firms traded 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. Nevertheless, because of full participation, the price is determined
by the average investor who has model uncertainty φ̄ for both factors A and B. With or
without merger, the average investor will hold the market portfolio. As a result, there is no
diversification discount.
5.2
Limited Participation
Now for the case of limited participation, the result is summarized in the following proposition.
Proposition 4 If (1 + ρ)xγσ < δ, there is limited participation in equilibrium and
P1 = P2 = (1 + ρ)[µ − σ(φ̄ − δ) − 2σ
γ(1 + ρ)xσδ],
PM = 2(1 + ρ){µ − σg −1[γ(1 + ρ)xσ]/2},
where
g(y) = (y − 2φ̄ + 2δ)3 − 2[(y − 2φ̄)+ ]3 /(48δ 2 ),
and PM < P1 + P2 , i.e., there is a diversification discount.
20
This proposition states that, with limited participation, the price of a conglomerate is
less than the sum of the prices of its components traded separately. Intuitively, when an
investor has low uncertainty on factor A, he is likely to participate in the market for factor
A portfolio when there is separate trading. However, the same investor may not have a low
uncertainty on factor B and may choose to stay away from factor B portfolio under separate
trading. When the two stocks are combined and traded as a bundle only, the investor can
no longer trade the stock for which he has a low uncertainty only and has to buy both factor
portfolios with equal weight. If there is no diversification discount, i.e., PM = P1 + P2 , the
investor may choose not to hold the bundled asset. As a result, the price of the conglomerate
has to be reduced to attract investors.24
Corollary 1 Keeping the average uncertainty φ̄ fixed, when there is limited participation,
diversification discount increases with uncertainty dispersion (δ).
Proposition 4 and Corollary 1 together predict that there exists a conglomerate discount
when there is limited participation. Moreover, since uncertainty dispersion is positively correlated with the fraction of investors not participating in the stock markets, diversification
discount is positively correlated with the nonparticipation rate. While diversification discount for conglomerates has been well documented in the literature (Lang and Stulz (1992),
Berger and Ofek (1995), and Rajan, Servaes, Zingales (2000), Hite and Owers (1983), Miles
and Rosenfeld (1983) and Schipper and Smith (1983)), our study is the first to establish the
relation between diversification discount and limited participation.
While existing empirical studies have yet to provide any guidance on the relation between
diversification discount and limited participation, our theoretical prediction suggests some
possible empirical tests. For example, surveys can be conducted to collect information on
investors’ uncertainty measures and participation status by asking a randomly selected group
of investors their range of expected payoffs (returns) for stocks of both single segment firms
and conglomerates as well as the investors’ ownership status on these stocks. The survey
information will allow for empirical analysis on how the diversification discount and limited
24
Our results are obtained in a two-risky asset economy. It is straightforward to extend the analysis to a
multiple-risky asset economy. However, to match the magnitude of diversification discount documented in
existing empirical studies, some form of market imperfection such as short sale constraint may need to be
imposed.
21
participation are related to investor’s uncertainty on the mean payoffs of these assets. For
another example, if some existing databases contain information on companies which are
merged at some point in time and information on investors’ holdings of the stocks of these
companies, then the information would allow empirical researchers to explore the association
between diversification discount and limited participation.
In Figure 5 we plot the diversification discount as a function of the model uncertainty
dispersion among investors (δ). 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
participation and the diversification discount is zero. As the uncertainty dispersion becomes
sufficiently large, limited participation occurs and the diversification discount becomes positive. As uncertainty dispersion increases, the diversification discount also increases. Both
panels in the figure suggest that the discount can be quite sizable.
Corollary 2 When there is limited participation, a conglomerate merger will further reduce
participation.
Figure 6 plots the nonparticipation rate for both the case where the two stocks are traded
separately (solid line) and the case in which the two stocks are merged into one conglomerate
(dashed line). The figure illustrates that at low levels of uncertainty dispersion, we observe
full participation. When the uncertainty dispersion becomes sufficiently large, investors with
high uncertainty optimally choose to stay sidelined. As the uncertainty dispersion 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 the two individual stocks separately.
6
Generalization
In this section we extend the model in Section 5 to allow for a more general factor structure
and demonstrate that our basic results are robust in the more general setting. Specifically,
22
the payoffs of stocks 1 and 2 are now given by
r1
r2
=β
where
β=
fA
fB
1
2
+
β1A β2A
β1B β2B
,
(21)
.
(22)
The matrix β represents the factor loadings and is assumed to be invertible. The factors fA
and fB are the systematic factors as before. The random shock 1 and 2 are the idiosyncratic
factor for stock 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 5,
investors do not have perfect knowledge on the stock payoffs. They know that the factors
(fA , fB , 1 , 2 ) follow a joint normal distribution. The risk of stock payoffs is summarized by
the non-degenerate variance-covariance matrix ΩF . We assume that investors have a precise
estimate of ΩF . Without loss of generality, we assume that fA and fB have equal variance
σ 2 , 1 and 2 have variance σ12 and σ22 , respectively. However, investors do not know exactly
the mean of fA or fB .25 Investors’ uncertainty is represented by the constraints
2
viA
σ −2 ≤ φ2iA ,
2
viB
σ −2 ≤ φ2iB .
Similar to Section 5, investors’ heterogeneity is characterized by φi = (φiA , φiB ) . We
assume that φiA and φiB have a joint distribution defined on S3 ≡ [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 ) . Let
x1 and x2 be the supplies of stock 1 and 2, respectively. The corresponding factor portfolio
supplies are given by
x≡
25
xA
xB
=β
By assumption, the idiosyncratic factors have zero mean.
23
x1
x2
.
6.1
Portfolio Choice and Market Participation
Under 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 σ 2 + (β1B Di1 + β2B Di2 )2 σ 2 + Di1
σ1 + Di2
σ2 .
2
(23)
The first line in Equation (23) represents the expected payoffs adjusted for uncertainty
for investor i. The second line derives from investors’ risk aversion. The demand for factor
portfolios are given by:
Di1
DiA
≡β
.
(24)
DiB
Di2
Since the matrix β is invertible, there is a one-to-one correspondence between the holdings
of factor portfolios and those of the stocks 1 and 2. In particular, if an investor does not
participate in the factor portfolio markets, he does not participate in the stock markets.
The first–order condition for investor i’s optimal holdings of the factor portfolios is given
by
µA − φiA σ + λiA
µB − φiB σ + λiB
−P = γ
σ2 0
0 σ2
+ (β
−1 )
σ12 0
0 σ22
β
−1
DiA
DiB
,
where λiA and λiB are nonnegative numbers such that 0 ≤ λiA ≤ 2σφiA and 0 ≤ λiB ≤ 2σφiB ,
and P ≡ (PA , PB ) = (β −1) (P1 , P2 ) . These constraints imply that when the investor holds
a long position in factor A, investor i takes the lowest expectation in his payoff distribution set
(λiA = 0). Similarly, if investor i holds a short position, he takes the highest expectation in his
payoff distributions set and thus λiA = 2σφiA . When he does not hold any position in factor
A, he takes the expectation between the two extremes, and in this case, 0 < λiA < 2σφiA .
Similar arguments hold for the constraint on λiB .
Let Di = (DiA , DiB ) , λi = (λiA , λiB ) , it follows that the demand for factor portfolios
is given by
Di = (γ Ω̂)−1 (µ − σφi + λi − P ) ,
24
(25)
where
Ω̂ =
σ2 0
0 σ2
+ (β
−1 )
σ12 0
0 σ22
β −1
is the variance-covariance matrix of the factor portfolios. Without loss of generality, we
assume that Ω̂ is positive.26 The demand for factor portfolios can be rewritten as
Di = (γ Ω̂)−1 [µ − P ] − (γ Ω̂)−1 [σφi − λi ] .
(26)
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. Given the
demand for the factor portfolios, the demand for individual stocks can be determined by
(Di1 , Di2 ) = β −1 Di .
(27)
We summarize the result on participation in the general setting as follows.
Theorem 1 If min{σφuA , σφuB } > M ≡ max{|µA − PA |, |µB − PB |}, then investors with
uncertainty level in S = [M, φuA ] × [M, φuB ] will take no position in the stock markets.
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 strictly positive, the equilibrium is unique. Moreover, both factor portfolios are traded at
a discount, i.e., µA − PA > 0 and µB − PB > 0.
Note that when the supply of a factor portfolio is zero, there could exist multiple equilibrium prices for that factor. Multiple equilibria occur because there could exist a continuum of
prices for that factor portfolio such that investors choose not to participate at these prices.
The aggregate demand for the factor portfolio is thus zero and the market for the factor
portfolio clears.
26
Otherwise, we can redefine a new set of factors fA = −fA , fB = fB such that Ω̂ is positive.
25
6.3
Equity Premium under Limited Participation
In equilibrium, the asset markets must clear, or equivalently, the factor portfolio markets
must clear, i.e.,
xA =
DiA =0,φi ∈S3
DiA h(φiA , φiB )dφiA dφiB ,
(28)
DiB h(φiA , φiB )dφiA dφiB .
(29)
xB =
DiB =0,φi ∈S3
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 (γ Ω̂)−1 .
(a) If γx ≥ σδ(ω11 − ω12 , ω22 − ω12 ) , then all investors hold both factor A and factor B
portfolios. In addition, the full participation equilibrium prices for the factor portfolios,
P f ≡ (PAf , PBf ) , are given by
P f = µ − σ φ̄ − γ Ω̂x.
(30)
(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 );
(c) If there is limited participation in both factor portfolios, PA > PAf and PB > PBf .
Market participation clearly depends on the dispersion of investors’ uncertainty. When
investors are relatively homogeneous, φuA − φlA and φuB − φlB are small, a full participation
equilibrium prevails. When investors are 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’ uncertainty is very heterogeneous with respect to both
factors, some investors do not participate in either factor portfolio and we have limited
participation in both markets.
26
6.4
Diversification Discount
In this subsection, we show that our results on the diversification discount still go through in
the general setting. Let the two firms merge to form a conglomerate. The payoffs from the
conglomerate is now rM = x1 r1 + x2 r2 . Without loss of generality, we normalize the share
supply of the conglomerate to one. We summarize the results in the following theorem.
Theorem 4 When there is full participation, there is no diversification discount, i.e., PM =
x1 P1 + x2 P2 . When there is limited participation, there is diversification discount, i.e., PM ≤
x1 P1 +x2 P2 . Moreover, the diversification discount is strictly positive when a positive measure
of investors participate only in one of the factor portfolios.
7
Conclusion
We have studied the relation among three well-documented empirical phenomena: limited
participation, the equity premium, and the diversification discount in a general equilibrium framework with model uncertainty. We find that the phenomena are fundamentally
connected in the heterogeneity of investors’ uncertainty about the distribution of stock payoffs.27 Specifically, the presence of model uncertainty leads to higher equity premium than
that obtained in standard expected utility framework without model uncertainty. Large dispersion in model uncertainty among investors results in limited participation. Merged firms
are traded at a discount when limited participation occurs in equilibrium.28 Diversification
discount can then cause further decrease in market participation.
Interestingly, we find that when there is limited market participation, caused by high
dispersion in uncertainty, equity premium can be lower than that in an economy with full
market participation. This is in contrast to the understanding in the literature that limited
participation tends to lead to higher equity premium.
27
It is conceivable that other forms of heterogeneity such as disappointment aversion, fixed pariticipation
cost also can generate similar results.
28
More generally, our results imply that sufficient dispersion in uncertainty makes packaging of securities
matter. Thus our model can be used to explain the close-end fund discount when there are short-sale
constraints and why tracking stocks exist.
27
With respect to the diversification discount, our model predicts that diversification discount and spin-off premium are larger when there is more uncertainty associated with the
stocks and investors are more heterogeneous. Such prediction is novel and can be empirically
tested in future research.
Overall, our study provides an integrated framework which allows us to analyze limited
participation, the equity premium, and the diversification discount simultaneouly. Our results suggest that model uncertainty and investors aversion to uncertainty are important
factors in understanding some of the well-documented asset pricing anomalies.
28
Appendix
Proof of Proposition 3: The proof for the case without merger is very similar to the single
stock case. Let x̂ = (1 + ρ)x. When γ x̂σ ≥ δ, there is full market participation. The prices
for the factor portfolios are given by
PA = PB = µ − φ̄ − γ x̂σ 2 .
As a result we have
P1 = PA + ρPB = PB + ρPA = P2 = (1 + ρ)[µ − φ̄ − γ x̂σ 2 ].
In the economy with merger, the maximization problem is
2
max DiM [2(1 + ρ)µ − (1 + ρ)sgn(DiM )(φiA + φiB )σ − PM ] − γDiM
(1 + ρ)2 σ 2 .
DiM
When γ x̂σ ≥ δ, we have
DiM =
2(1 + ρ)µ − (1 + ρ)(φiA + φiB )σ − PM
.
2(1 + ρ)2 γσ 2
Again there is full market participation. Aggregating the demands across all investors and
equal the aggregate demand to aggregate supply, we arrive at
x=
2(1 + ρ)µ − 2(1 + ρ)φ̄σ − PM
,
2(1 + ρ)2 γσ 2
which implies
PM = 2(1 + ρ)[µ − φ̄ − γ x̂σ 2 ] = (1 + ρ)(PA + PB ) = P1 + P2 ,
and there is no diversification discount.
Proof of Proposition 4: When γ x̂σ < δ, in the economy without merger, there is limited
participation. Following the results in Proposition 2, we get
PA = PB = µ − σ(φ̄ − δ) − 2σ
29
γ x̂δσ.
As a result we have
P1 = PA + ρPB = PB + ρPA = P2 = (1 + ρ)[µ − σ(φ̄ − δ) − 2σ
γ x̂δσ].
(A1)
When γ x̂σ < δ, in the economy with merger, there is also limited participation. Thus,
there exists a φ̂ such that when φiA + φiB ≥ φ̂, investor i will not invest in the conglomerate.
φ̂ is determined by,
[2(1 + ρ)µ − (1 + ρ)φ̂σ − PM ] = 0.
(A2)
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,φi ∈S2
1
[2(1 + ρ)µ − (1 + ρ)(φiA + φiB )σ − PM ]dφiA dφiB .
8(1 + ρ)2 γσ 2 δ 2
Utilizing Equation(A2), we arrive at the following market clearing condition
(1 + ρ)γσx =
φiA +φiB <φ̂,φi ∈S2
[φ̂ − (φiA + φiB )]
dφiA dφiB = g(φ̂),
8δ 2
where
g(φ̂) ≡
(φ̂ − 2φ̄ + 2δ)3 − 2[(φ̂ − 2φ̄)+ ]3
.
48δ 2
(A3)
We claim that g is an increasing function of φ̂. To show this, define z = φ̂ − 2φ̄. Then we
must have z ≤ 2φ̄ + 2δ − 2φ̄ = 2δ. It is clear that g is an increasing function in φ̂ when
z ≤ 0. When 2δ ≥ z > 0,
dg
2δ 2 − (z − δ)2
=
> 0.
dz
16δ 2
Thus, g is also strictly increasing in φ̂ when 2δ ≥ z > 0. The equilibrium price of the
conglomerate is thus given by
PM = 2(1 + ρ)[µ − σg −1 (γ x̂σ)/2].
30
(A4)
It is readily verified that PM < P1 + P2 . Later in the proof of Theorem 4, we provide the
proof for a more general case.
Proof of Corollary 1: We prove the case for ρ = 0, γ = 1, x = 1, σ = 1. The general
case can be dealt with similarly. When δ = 6, we have from the proof of Proposition 4 that
g(φ̂) = 1 = δ/6 = g(2φ̄). When δ ≥ 6, since g(·) is an increasing function, we must have
√
3
φ̂ ≤ 2φ̄. Consequently, g(φ̂) = 1 implies that φ̂ = 2 6δ 2 + 2(φ̄ − δ). By the expressions for
√
√
3
Pi (Equation (A1)) and PM (Equation (A4)), the diversification discount is 2( 6δ 2 − 2 δ),
which is increasing in δ.
In the case in which 1 < δ < 6, we have φ̂ > 2φ̄. The diversification discount is
√
z = φ̂ − 2φ̄ + 2δ − 4 δ. Substituting this back into g(φ̂) = 1, we have
√
√
F (z, δ) = (z + 4 δ)3 − 2(z − 2δ + 4 δ)3 − 48δ 2 = 0.
Define
ψ≡
√
δ,
ω≡
z + 4ψ
.
2ψ 2
Since 0 < φ̂ − 2φ̄ < 2δ, 2δ < z + 4ψ < 4δ and 1 < ω < 2. Substituting into F , we have
G(ω, ψ) ≡ ω 3 − 2(ω − 1)3 −
6
= 0.
ψ2
Since Gω (ω, ψ) = −3(ω − 2)2 + 6, Gω > 0 for 1 < ω < 2. Thus G(ω, ψ) = 0 implicitly defines
√
a function ψ → ω for 1 < ψ < 6 and 1 < ω < 2. From the definition of ω, we have
dω
Gψ
2
dz
= 4(ψω − 1) + 2ψ 2
= 4(ψω − 1) − 2ψ 2
].
= 4[ψω − 1 −
2
dψ
dψ
Gω
ψ(−ω + 4ω − 2)
With some algebraic derivation, the inequality dz/dψ > 0 is equivalent to
ω 3 − 2(ω − 1)3
>
ω−
3(−ω 2 + 4ω − 2)
ω 3 − 2(ω − 1)3
.
6
(A5)
ω 3 − 2(ω − 1)3 2 ω 3 − 2(ω − 1)3
] >
.
3(−ω 2 + 4ω − 2)
6
(A6)
From (A5) we get
[ω −
31
Using one to subtract both sides of (A6), we get
[1 − ω +
ω 3 − 2(ω − 1)3
ω 3 − 2(ω − 1)3
6 − ω 3 + 2(ω − 1)3
][1
+
ω
−
]
>
.
3(−ω 2 + 4ω − 2)
3(−ω 2 + 4ω − 2)
6
(A7)
Multiplying both sides of Equation (A7) by 18/(2 − ω), we get
a(ω)b(ω)
< d(ω),
c(ω)
(A8)
where
a(ω) = −2ω 3 + 3ω 2 + 12ω − 8,
b(ω) =
9
5
− (2ω − )2 ,
4
2
c(ω) = [2 − (2 − ω)2 ]2 ,
d(ω) = 3(6 − (2 − ω))2.
It is easy to verify that a, c, d are increasing functions of ω for 1 < ω < 2. The function b(ω)
is increasing for 1 < ω < 5/4 but decreasing for 5/4 ≤ ω < 2 and maximized at ω = 5/4
with value 9/4. Therefore, for 1 < ω ≤ 9/8 we have
a(9/8)b(9/8)
a(ω)b(ω)
<
< d(1) < d(ω),
c(ω)
c(1)
and, for 9/8 < ω ≤ 3/2 we have
a(3/2)b(5/4)
a(ω)b(ω)
<
< d(9/8) < d(ω).
c(ω)
c(9/8)
Finally, for 3/2 < ω ≤ 2, we have
a(2)b(3/2)
a(ω)b(ω)
<
< d(3/2) < d(ω).
c(ω)
c(3/2)
Consequently, z is an increasing function of ψ and thus an increasing function of δ.
32
Proof of Corollary 2: We prove the case for ρ = 0, γ = 1, x = 1, σ = 1. The more general
case can be treated similarly. First, consider δ ≥ 6. In this case φ̂ ≤ 2φ̄. From expression
(13), the proportion of nonparticipants is
φiA +φiB ≥φ̂,φi ∈S2
dφiA dφiB
φ∗ − φ̄ + δ 2
]
=
[1
−
=
(1
−
1/δ)2 ,
4δ 2
2δ
in the economy in which the two stocks are traded separately. For the economy with the
conglomerate, investors with φiA + φiB ≥ φ̂ will not participate. In this case, since δ ≥ 6,
from expression (A3), we have g(φ̂) = 1 ≥ 6/δ = g(2φ̄). Since g is an increasing function,
we must have φ̂ ≥ 2φ̄. The proportion of nonparticipants is
φiA +φiB ≥φ̂,φi ∈S2
(φ̂ − 2φ̄ + 2δ)2
dφiA dφiB
=
1
−
= 1−
4δ 2
8δ 2
where the second equality follows from expression (A3). Let x =
3
9
,
2δ 2
6
1/δ.
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 (A9) is an increasing function of x and when x =
(A9)
6
1/6, the expression holds.
Therefore the corollary holds for the case in which δ ≥ 6.
Next, we consider the case in which 1 < δ < 6. In this case, it can be similarly shown
that φ̂ < 2φ̄ since g(φ̂) = 1 < 6/δ = g(2φ̄) and g is an increasing function. In the economy
with the conglomerate, the proportion of nonparticipants is
φiA +φiB ≥φ̂,φi ∈S2
dφiA dφiB
(2φ̄ + 2δ − φ̂)2
=
.
4δ 2
8δ 2
We need to show that
φ∗ − φ̄ + δ 2
(2φ̄ + 2δ − φ̂)2
]
>
[1
−
=
(1
−
1/δ)2 ,
8δ 2
2δ
33
which reduces to
√
√
φ̂ < φ̌ ≡ 2φ̄ + 2δ − 2 2δ + 2 2δ.
To prove φ̂ < φ̌, notice that g(φ) is an increasing function, and we need to prove that
g(φ̌) > g(φ̂) = 1.
(A10)
Substituting the expression of φ̌ into inequality (A10), we get
√
√
√
√
(4δ − 2 2δ + 2 2δ)3 − 2(2δ − 2 2δ + 2 2δ)3
> 1.
48δ 2
Let y =
√
(A11)
δ, with some algebra, inequality (A11) reduces to
√
√
y 3 + 3( 2 + 1)y − (4 + 3 2) > 0,
(A12)
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 = 2σφiA if DiA < 0 and 0 ≤ λiA ≤ 2σφiA if DiA = 0. Left multiply both
sides of equation (26) by (µ − P + λi − φi σ) . Then the RHS is positive and the LHS is
negative. Thus µ − P + λi − φi σ must be zero and Di = 0.
Proof of Theorem 2: Notice that 0 ≤ λiB ≤ 2σφiB . For existence, define the aggregate
demand by the right side of equations (28) and (29) and denote it by D(P ). Using the
Maximum Theorem, it is readily shown that after the inner minimization, the objective
function in (23) 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) = [µ − σ φ̄ + λ(P )] − γ Ω̂x,
where
λ(P ) =
φi ∈S3
λi (P ) h(φiA , φiB )dφiA dφiB .
This mapping is continuous in P . Let
S = {[µ − σ φ̄ + λ] − γ Ω̂x, 0 ≤ λ ≤ 2σ(φ̄ + δ)(1, 1) }.
34
Clearly, M(S) ∈ S. Applying the Brouwer’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 .
Notice that ω12 is off-diagonal element of (γ Ω̂)−1 and is negative by assumption. Thus we
must have PB∗ > PB . We show that at price vector P ∗ = (PA∗ , PB∗ ) = (PA , PB∗ ) the demand for
∗
> xA = DA .
factor portfolio A will be less than the supply. Suppose, to the contrary, that DA
∗
∗
> DiA . It follows that λiA ≥ λ∗iA (This follows because DiA
> 0 implies
Then for some i, DiA
∗
∗
= 0 implies DiA < 0 which implies that λiA = 2σφiA ; and finally DiA
< 0
λ∗iA = 0; DiA
implies DiA < 0 which implies λiA = 2σφiA ≥ λ∗iA ). Noting that
∗
0 > DiA − DiA
= ω11 (PA∗ − PA + λA − λ∗iA ) + ω12 (PB∗ − PB + λiB − λ∗iB ),
(A13)
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.
(A14)
Noting that PA∗ > PA and the two terms in the brackets of Equation(A13) and Equation
(A14) are all positive, we have
ω21 ω12 > ω11 ω22 ,
which is a contradiction. Consequently, at price vector P ∗ = (PA∗ , PB∗ ) = (PA , PB∗ ) the
demand for factor portfolio A will be less than the supply.
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 >
35
PB∗ − PB = −ω11 (PA − PA )/ω12 > 0. Similarly, we can show that PA − PA > −ω22 (PB −
PB )/ω12 > 0. These two inequalities together violates the positive definiteness of Ω̂−1 . Thus
the equilibrium must be unique.
Next we show that µA > PA and µB > PB . Since the supply of the factor portfolios are
positive, there must be some investors who hold strictly long positions in factor portfolio A.
Let i be such an investor. There are several cases. Suppose first that investor i also holds a
strictly long position in factor portfolio B, we have
µ − P = γ Ω̂Di + σφi > 0.
Thus µA − PA > 0 and µB − PB > 0.
Secondly, suppose that investor i holds positive amount in factor portfolio A and zero in
factor portfolio B. Then the equity premium for A must be positive as
µA − PA = γ Ω̂11 DiA + σφiA > 0.
In the third scenario, suppose that investors who hold strictly long positions in factor
portfolio A will only hold short positions in factor portfolio B. It follows that investors who
hold long positions in factor portfolio B must hold non-positive positions in factor portfolio
A. Suppose that there exists an investor j who hold strictly long positions in factor portfolio
B but zero position in factor portfolio A, then we must have µB − PB = γ Ω̂22 DjB + φjB > 0.
Moreover, we have µA − PA > σf φiA > 0. Otherwise, we must have µA − PA − σf φiA ≤ 0,
and
(µA − PA − σf φiA , µB − PB + σf φiB ) Di =
(µA − PA − σf φiA , µB − PB + σf φiB ) (γ Ω̂)−1 (µA − PA − σf φiA , µB − PB + σf φiB > 0,
and the RHS is strictly positive while the LHS is clearly negative.
Since investor i in the three cases above is arbitrary, we are left with the case where
investors who hold strictly long positions in factor portfolio A hold short positions in factor
portfolio B, and investors who hold long positions in factor portfolio B hold short positions
in factor portfolio A. Let i be an investor who hold positive in A, negative in B and let j be
36
an investor who hold positive in B and negative in A. We have
(−φiA − φjA , φiB + φjB )(Di − Dj ) = (−φiA − φjA , φiB + φjB )(γ Ω̂)−1 σ(−φiA − φjA , φiB + φjB ) .
The LHS is negative while RHS is strictly positive. This is a contradiction. Therefore the
last case is impossible. This finishes the proof that µA − P > 0. Similarly, it can be shown
that µB − PB > 0.
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,
without loss of generality, that for a positive measure of investors, the demand for factor
portfolio A is negative and the demand for factor portfolio B is nonzero. Fix such an investor.
Let his uncertainty about factor portfolio A be given by φiA . Let φB = sup{φjB , DjA < 0}.
Then,
D(φiA , φB ) = ω11 (µA − PA − σφiA + λiA ) + ω12 (µB − PB − σφB ) ≤ 0.
Thus for all φjB > φiB , we have DjA > 0, and for all φjB < φiB , we have DjA < 0 because
ω12 < 0. Thus λjA = 0 for φjB > φB and λjA = 2σφjA , for φjB < φB . This implies that λiA
is not continuous in φB . This is a contradiction.
Using the result just proved, it can be verified that, if investors fully participate, the
equilibrium prices for the factor portfolios are given by (30). Given positive supply and
the conjectured equilibrium price, it can also be verified that investors have strictly positive
demand (equation (26)) for both factor portfolios and markets clear. This proves part (a).
For (b) and (c), suppose that some investors do not participate in at least one of the
factor portfolios. Multiplying (26) by γ Ω̂, integrating and using (30), we get
f
P −P =λ ≡
φi ∈S3
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 statements in the proposition hold.
37
Proof of Theorem 4: Let x = (xA , xB ) , P = (PA , PB ) . Let
rA
rB
= β
−1
r1
r2
,
which is the payoff vector of the factor portfolios. Clearly, x1 r1 + x2 r2 = xA rA + xB rB
and x1 P1 + x2 P2 = xA PA + xB PB . So we simply view the merged firm as having payoff
xA rA + xB rB and give the proof in terms of factor portfolios.
Denote µM ≡ x µ. We deal with the full participation case first. When the firms are
merged,
1=
φi ∈S3
DiM dφiA dφiB =
φi ∈S3
f
(x Ω̂x)−1 [µM − PM
− σx φi + λiM ]dφiA dφiB .
As in the case of single stock, λiM = 0 in equilibrium. When the two stocks are traded
separately, as shown in the proof of Theorem 3, λiA = λiB = 0, and hence
γ Ω̂x = µ − P − σ φ̄,
γx Ω̂x = µM − PM − σx φ̄.
It is easy to check that our equilibrium demand satisfies the market clearing condition.
λM
f
+ λM , where
With limited market participation, we have P = P f + λ and PM = PM
≡ φi ∈S3 λiM h(φiA , φiB )dφiA dφiB . We will show that at price P̂M = x P the market
demand for the bundled stock is less than one. Since the demand curve is downward sloping,
in equilibrium, PM < P̂M = x P .
We first prove that for all investors, the following condition holds:
λiM ≤ x λi .
(A16)
First of all, expression (A16) must hold when the demand for the merged firm is positive
for investor i as λiM = 0. Now consider the case where investor i holds zero positions in
both factor portfolios. In this case, the investor must hold zero positions in the merged firm
and
0 = x (γ Ω̂)Di = µM − P̂M − σx φi + x λi = µM − P̂M − σx φi + λiM .
38
Thus λiM = x λi , and condition (A16) holds too.
We are left with cases in which investors hold zero positions in the merged firm but
nonzero positions when the stocks are traded separately. In the economy without the merger,
since the excess payoffs of the two factor portfolios are both positive (Theorem 2), it is
impossible for any investor to hold negative positions in both factor portfolios or zero in
one factor portfolio and negative in the other (recall that each component of Ω̂ is positive).
Moreover, it is not possible for investor i’s optimal demand for the two factor portfolios to
be both positive. Otherwise, we have
f
− σx φi + λiM ] > 0,
DiM = (x Ω̂x)−1 [µM − PM
which contradicts with the assumption that DiM = 0. Thus, the remaining case is that
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 factor portfolio B is positive.
Then, we must have µB − PB − σφiB > 0. Otherwise, multiplying both sides of equation (26)
by µ − P − σφi + λi would lead to a contradiction. Now suppose that investor i’s demand
for factor portfolio A is negative. Then λiB = 0, λiA = 2σφiA . Since DiM = 0, we have
λiM = −x (µ − P − σφi ) < xA (σφiA − (µA − PA )) < xA σφiA < 2σφiA xA = x λi . Finally, if
investor i holds zero position in factor portfolio A, then
γ Ω̂12 DiB = µA − PA − σφiA + λiA ,
γ Ω̂22 DiB = µB − PB − σφiB .
Dividing yields λiA = −(µA − PA − σφiA ) + Ω̂12 (µB − PB − σφiB )/Ω̂22 . Hence
xA λiA + xiB λiB = −xA (µA − PA − σφiA ) + xA Ω̂12 (µB − PB − σφiB )/Ω̂22
> −xA (µA − PA − σφiA ) > −x (µ − P − σφi ) = λiM .
To summarize, condition (A16) holds for all cases. Consequently, the aggregate demand
39
for the merged firm at price P̂M is:
DM =
φi ∈S3
DiM h(φiA , φiB )dφiA dφiB = (x Ω̂x)−1 [x (µ − P − σ φ̄) + λM )
≤ (x Ω̂x)−1 [x (µ − P − σ φ̄ + λ)
= (x Ω̂x)−1 (x Ω̂x) = 1.
(A17)
Thus when there is limited participation, the market demand is less than 1 for the conglomerate at price P̂M = x P . For the market to clear, the price of the conglomerate must be
larger than x P . Moreover, the inequality holds strictly when a positive measure of investors
holds long positions in one factor portfolio but not the other.
40
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44
Figure 1: Asset Price under Limited Market Participation
1.3
1.25
1.25
1.2
1.15
Price
Price
1.2
1.1
1.05
1.15
1.1
1
0.95
0
0.2
0.4
0.6
0.8
Average model uncertainty
1.05
1
1.25
0
0.1
0.2
0.3
Supply of asset
0.4
1.2
1.2
1.15
Price
Price
1.15
1.1
1.1
1.05
1.05
1
0.95
0
1
2
3
Risk aversion
4
1
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 (δ). The baseline parameter values are set at
x̄ = 0.25, µ = 1.2, σ = 0.3, γ = 1, φ̄ = 0.5, and δ = 0.4.
45
Figure 2: Proportion of Nonparticipation in the Asset Markets
1
Proportion of nonparticipation
Proportion of nonparticipation
0.75
0.7
0.65
0.6
0.55
0.5
0.1
0.2
0.3
0.4
Asset payoff volatility
0.7
0.6
0.5
0
0.1
0.2
0.3
Supply of asset
0.4
0.8
Proportion of nonparticipation
Proportion of nonparticipation
0.8
0.4
0.5
1
0.8
0.6
0.4
0.2
0
0.9
0
1
2
3
Risk aversion
4
0.6
0.4
0.2
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 (σ), stock supply (x), risk aversion (γ), and model uncertainty dispersion (δ). The
baseline parameter values are set at x̄ = 0.25, µ = 1.2, σ = 0.3, γ = 1, φ̄ = 0.5, and δ = 0.4.
46
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H.+(91(
20
Direct participation
Total participation
18
Annual stock returns (%)
16
14
12
10
8
6
4
5
10
15
20
25
30
35
Stock market participation (%)
40
45
50
55
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0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
Supply of stock
0.1
0.05
0.2
0.3
0.4
Payoff volatility
0.5
0.2
0.3
Premium decomposition
Premium decomposition
0.15
0
0.1
0.4
0.35
0.25
0.2
0.15
0.1
0.05
0
0.2
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
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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
\ 05|3911054)6/.0$'3%(0$1)'3+(-›0$
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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
›'3(79:.0$)0$7/.0$'3Ek05.('3+Zj^†1'3#$0$%mb"(%
k05.€^†%1(%mb,91239k›u<†+(()H.0$'3
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