Model Uncertainty, Limited Market Participation and Asset Prices

Transcription

1 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 , USA, (919) , (919) ; and Tan Wang is with the Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, British Columbia, Canada, V6T 1Z2, (604) 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.

2 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.

3 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

4 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

5 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

6 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

7 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

8 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 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

9 agent s preference is represented by a multi-prior expected utility (Gilboa and Schmeidler (1989)) { min E Q [u(w )] }, (1) Q P where E Q 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% 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

10 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) wherev denotes the adjustment to the estimated mean stock payoff and satisfies: v 2 σ 2 φ 2, (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 interval 18 S 1 =[ φ δ, φ + δ], where φ δ and δ measures the dispersion of the uncertainty among investors. When φ = 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

11 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 W 1i = W 0i + D i (r P ), where W 0i and W 1i are investor i s beginning of period and end of period wealth, respectively, D i 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 D i v D i (µ v P ) γσ2 2 D2 i, 9

12 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 max D i D i [µ sgn(d i )φ i σ P ] γσ2 2 D2 i, (3) where sgn( ) is an indicator function which takes the sign of its argument. The expression, µ sgn(d i )φ i σ, represents the uncertainty-adjusted expected payoff under the worst case scenario for an investment strategy of D i shares in the stock. The intuition behind (3) is the following. If investor i takes a long position (D i > 0andsgn(D i ) = 1), the worst case scenario for him is that the mean excess payoff is given by µ φ i σ P. Similarly, if he takes a short position (D i < 0andsgn(D i )= 1), the worst case scenario is that the mean excess payoff is µ + φ i σ P. The investor s optimal holding in the stock is given by 1 (µ φ γσ 2 i σ P ) if µ P>φ i σ, D i = 0 if φ i σ µ P φ i σ, 1 (µ + φ γσ 2 i σ P ) if µ P< φ i σ. (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 = φ i σ (v = φ i σ). When the price P falls in the region, [µ φ i σ, µ + φ i 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

13 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 (µ σφ i P ) 1 x = φ i S 1 γσ 2 2δ dφ i = 1 γσ (µ σ φ P ). 2 Solving for P, we arrive at the following equilibrium price for the stock, P = µ γσ 2 x σ φ. (5) As long as there exists model uncertainty ( φ >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 φ. (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 ( φ) 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 ( φ + 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. (7) 11

14 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 = Solving for φ yields D i φ i <φ,φ i S 1 2δ dφ i = 1 (φ φ + δ) γσ 2 2δ [ µ σ(φ + φ ] δ) P. (9) 2 φ = φ δ +2 γδσx. (10) Using Equation (8), we arrive at the following equilibrium price for the stock: P = µ σ( φ δ) 2σ γδσx. (11) Once again, in the presence of model uncertainty, the stock is traded at a discount relative to its expected payoff Note that φ δ 0 by definition. 12

15 To gain intuition of this pricing formula, we rewrite Equation (9) as µ P = γσ2 x α + φ pσ, (12) where α = φ ( φ δ) 2δ = γσx δ, (13) is the proportion of investors holding long positions in the stock, and φ p = 1 2 (φ + φ δ) = φ δ + γδσx, (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 (φ p ) is lower than the unconditional average uncertainty ( φ). 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 theriskpremium. Thisisconsistentwithwhat 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 ( φ), 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

16 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

17 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., φ p / 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

18 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 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). Theriskpremiumis unrelated to the average model uncertainty ( φ) but increases with the uncertainty dispersion (δ). However, the uncertainty premium increases with the average model uncertainty ( φ) 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

19 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 (φ p ), 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 r 1 = f A + ρf B, (15) r 2 = ρf A + f B, (16) 17

20 where r 1 and r 2 are payoffs on stock 1 and 2, respectively, f A and f B 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 x 1 and x 2,withx 1 = x 2 = x>0. All variables are jointly normally distributed. Investors know that the factors, f A and f B, follow a joint normal distribution with a non-degenerate variance-covariance matrix Ω F 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 (Ω F ). Without loss of generality, we assume that Ω F is diagonal with common element σ 2. However, investors do not know exactly the mean of f A or f B.Letµ A = µ B = µ be the reference level for the mean of f A and f B.Letv A and v B be the adjustment to the mean µ A and µ B, respectively. We therefore define the set P by the following constraints on v A and v B : va 2 σ 2 φ 2 A, (17) and v 2 B σ 2 φ 2 B, (18) where φ A and φ B 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 f A and f B fall into the set {(µ A + v A,µ B + v B ): v A and v B satisfy (17) and (18), respectively}. We assume that φ =(φ A,φ B ) are uniformly distributed on the square 22 where φ δ. S 2 =[ φ δ, φ + δ] [ φ δ, φ + 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 r M = r 1 + r 2 =(1+ρ)(f A + f B ). 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

21 For ease of exposition, we introduce factor portfolios tracking the two factors A and B. Let D ia and D ib be investor i s holdings of the factor portfolio A and B, respectively. Let P A and P B be the prices of factor portfolio A and B, respectively. Let P j denote the price of the stock j, D ij denote investor i s demand for stock j, forj =1, 2. Let D im be investor i s demand for the merged firm and P M 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: f A r 1 ρr 2 1 ρ 2, f B r 2 ρr 1 1 ρ 2, and the demand for the factor portfolios, D ia and D ib, can be written as: D ia = D i1 + ρd i2, D ib = ρd i1 + D i2. Similarly, we can rewrite the demand for the stocks in terms of the demands for the factor portfolios: D i1 = D ia ρd ib 1 ρ 2, D i2 = D ib ρd ia 1 ρ 2. The supply of each factor portfolio is ˆx (1 + ρ)x. Since the two factors are orthogonal to each other and investors have CARA utility function, the prices of the factors (P A and P B ) 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 P 1 = P A + ρp B and P 2 = ρp A + P B. 5.1 Full Participation We first consider the case of full participation. The following proposition summarizes the main results under full participation. 19

22 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: P 1 = P 2 =(1+ρ)[µ σ φ γ(1 + ρ)xσ 2 ]. (19) and the price for the conglomerate in the economy after the merger is given by P M = P 1 + P 2 =2(1+ρ)[µ σ φ 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., P M = P 1 + P 2. 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 φ 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 P 1 = P 2 =(1+ρ)[µ σ( φ δ) 2σ γ(1 + ρ)xσδ], P M =2(1+ρ){µ σg 1 [γ(1 + ρ)xσ]/2}, where g(y) = { (y 2 φ +2δ) 3 2[(y 2 φ) + ] 3} /(48δ 2 ), and P M <P 1 + P 2, i.e., there is a diversification discount. 20

23 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., P M = P 1 + P 2,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 φ 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