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UBC Theses and Dissertations

Heterogeneity in financial markets Rubin, Amir 2003

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HETEROGENEITY  IN FINANCIAL MARKETS by  Amir Rubin  B.A., The Hebrew U n i v e r s i t y o f Jerusalem, M.A., The Hebrew U n i v e r s i t y o f Jerusalem,  1993 1995  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Sauder  School o f Business; Department o f Finance) We accept t h i s t h e s i s as conforming to they^^rj^ired s t a n d a r d  THE UNIVERSITY OF BRITISH COLUMBIA August 2003 © Amir Rubin,  2003  In  presenting  degree  at  the  freely available copying  of  department  this  University of  publication of  partial  fulfilment  of  the  his  or  her  representatives.  for  It  is  granted  by the  understood  this thesis for financial gain shall not be allowed  of  £;r>(x.n<Lc . 9  •  So-^  The University of British Columbia Vancouver, Canada  Date  DE-6  (2/88)  c^0  03  €>r  &ck<*>l  advanced  Library shall make it  that  extensive  head of copying  my or  without my written  permission.  Department  an  agree that permission for  thesis for scholarly purposes may be by  requirements  British Columbia, I agree that the  for reference and study. I further  this or  thesis in  iSu^'nes^  Abstract The thesis studies different forms of heterogeneity and their effect on financial markets. The first chapter deals with a conflict that is typically neglected in the corporate finance literature. Shareholders want to maximize their portfolio value and not any specific firm value. These two objectives are quite different if firms affect the cashflow of other firms in the market. We show that undiversified corporate managers who are compensated according to firm value tend to pursue projects that increase firm value but not necessarily portfolio value. This conflict can be reduced by issuing debt, and can be reduced even more with option grants. Both contracts create a risk-shifting effect, which offsets the incentive of the manager to engage in some safe projects that do not enhance the opportunity set. We show that promoting risk-shifting while incurring some under-investment, induces managers to take on comparatively more opportunity set enhancing projects, which in turn increase the shareholder's portfolio value. The second chapter deals with a welfare comparison between the competitive limit order book structure and the monopolist market maker structure. The comparison is done in an economy where market volume increases liquidity. We show that in a symmetric information economy, where investors differ only in their initial endowments, a market maker can solve a free-rider problem that exists in the competitive limit order book market. Since individual investors assume that their own trade has no effect on market liquidity, they disregard it when maximizing their expected utility. A monopolist market maker that provides a pricing schedule to the market and exerts search and promotion costs to increase trading volume can solve the free-rider problem. The market maker facilitates all trade in the economy and guarantees a reduction in liquidity risk. Under some circumstances, the benefits of solving the free-rider problem outweigh the costs of having a market maker who collects a fee. When allowing for competition, we have a free-rider problem from the market makers perspective. As more market makers compete in the market, each market maker can commit to relatively lower total volume, which results in a double free-rider problem. Thus, while a monopolist market maker can endogenize the volume decision completely, a competitive structure cannot. Finally, the third chapter deals with the removal of short sale constraints when investors have different beliefs about the final payoff distribution of a security. Our focus is the effect on the price volatility of the underlying asset. The intuition is derived from simple geometry. We show that the price curve as a function of the uncertain future payoff changes when investors are able to act on the belief that the price of the share is relatively high. In a very simple model with successive generations of singleperiod investors, we show that volatility can either increase or decrease, depending on ii  the variability of news about final payoffs. As an empirical illustration, we consider data from the Israeli stock market. The data show that volatility increased following the initiation of index options, consistent with the fact that short sales were prohibited in Israel when index options were introduced.  iii  TABLE OF CONTENTS  ABSTRACT  ii  LIST O F FIGURES  vii  ACKNOWLEDGMENTS  viii  CHAPTER I  WHAT DO SHAREHOLDERS REALLY WANT? INCENT I V E C O N T R A C T S A N D D E B T P O L I C Y W H E N INVESTORS A R EDIVERSIFIED  1  1.1  Introduction  1  1.2  A l l Equity Firm  6  1.3  A Leveraged F i r m  10  1.4  O p t i o n Grants to Manager  18  1.5  E m p i r i c a l Implications  22  1.6  Conclusion  25  REFERENCES CITED C H A P T E R II  27  A RATIONALE FOR THE MARKET MAKER TURE  STRUC50  2.1  Introduction  50  2.2  The Model  54  2.3  L i m i t Order Book Versus Market M a k e r  63  2.4  C o m p e t i t i o n between M a r k e t Makers  67  2.5  Discussion  71  2.6  Concluding Remarks  75  REFERENCES CITED  77 iv  C H A P T E R III  T H E E F F E C T OF SHORT S A L E CONSTRAINT R E M O V A L ON VOLATILITY IN T H E P R E S E N C E O F H E T E R O G E NEOUS BELIEFS 92  3.1  Introduction  92  3.2  The Model  94  3.3  V o l a t i l i t y Effects  99  3.4  E m p i r i c a l Illustration  103  3.5  Summary and Conclusions  107  REFERENCES CITED  109  LIST O F F I G U R E S Figure Ll  Page  T i m i n g of the debt model: A t t = 0 diversified shareholders issue a face value D of risky debt. T h i s debt is valued i n the financial markets at a price of pa and the proceeds of the debt issue are distributed to shareholders as dividends. A t t — 1, investors choose the fixed wage WQ and the incentive share a that they give to the manager. A t t = 2, the manager of firm A decides on the two types of effort E and E . A t t = 3, the cashflows of the firms i n the economy is generated and the payoffs are distributed to the manager, shareholders, and debtholders Debt model: the t = 3 cashflow distribution to manager, shareholders, and debtholders i n the 2 possible states, bankruptcy (probability fe) and solvency (with probability 1 — fe) T h e relative advantage of having debt-in-place compared to an a l l equity firm for different levels of cannibalistic effort payoff c and economy increasing effort payoff i n the low (bankruptcy) state si. T h e numeraire is the high state economy increasing effort payoff s = l A 3-dimensional illustration of the relative advantage of having debtin-place compared to an all equity firm for different levels of cannibalistic effort payoff c and economy increasing effort payoff i n the low (bankruptcy) state s/,. T h e numeraire is the high state economy increasing effort payoff = 1 O p t i o n grants model: the t = 3 cashflow distribution to manager and shareholders, i n the two possible states, the down state (probabili t y b) and up state (with probability 1 — fe) T h e relative advantage of having option grants compared to an a l l equity firm for different levels of cannibalistic effort payoff c and economy increasing effort payoff i n the low (bankruptcy) state s/.. T h e numeraire is the high state economy increasing effort payoff s = 1 c  1.2  1.3  s  h  1.4  1.5  1.6  h  T h e commitment cost: the difference between the advantage from options grants and the advantage from debt-in-place represents the commitment cost. T h e commitment cost increases w i t h the magnitude of cannibalistic effort payoff c/sh II. 1 A schematic illustration of the feedback loop between volume and liquidity: Vo— volume at time t = 0; A, B — time t = 1 expected price of the stock; a A, O- — variance of good and bad state prices at t = 1 '  43  44  45  46  47  48  1.7  49  B  vi  83  11.2 T h e Edgeworth Trade B o x and the formulation of volume and price depending on the initial endowments of the investors 11.3 A schematic illustration of the derivation of the market maker pricing schedule: S(Pi)— the supply curve of excess volume that the market maker provides the market for a given P*; MR(Pi)— the marginal revenues curves derived from investors demand for excess volume for a given price Pi 11.4 T h e equilibrium price of the risky asset i n the limit order book structure and the market maker structure II. 5 Volume i n the limit order book strucutre and the market maker structure II. 6 Fee i n the market maker structure II. 7 Welfare comparison between the limit order book trade structure and the market maker trade structure. . 11.8 T h e Certainty Equivalent (CE) for initial stock endowments of Xi = 0.9, X = 0.1 and different cash endowments (x-axis), where Ci —  84  85 86 87 88 89  2  90 11.9 Welfare comparison between the limit order book trade and the market maker structure for different degrees of competition among market makers, m is the number of market makers. (Black region - market maker structure optimal. W h i t e region - limit order book structure optimal) I I I . l Scehmatic description of the two aggregate investors' beliefs about future stock payoff (no > n ). T h e price i n a constrained economy is P , while the price i n an unconstrained economy is P " III. 2 Schematic description of the price as a function of the A-values i n the constrained and unconstrained markets. T h e P curve is for the constrained economy, where as the P curve is for the unconstrained market, where options are traded 111.3 Schematic description of the price as a function of the P-values i n the constrained and unconstrained markets. T h e P curve is for the constrained economy, where as the P curve is for the unconstrained market, where options are traded 111.4 T w o cases of increasing volatility. Panel A illustrates a case where AA > AB and the difference A\ — B is large. Panel B illustrates a case where A A < AB and the difference A\ — B is small. . . .  91  p  Ill  u  112  u  113  2  2  vii  114  ACKNOWLEDGMENTS The completion of this thesis has been a long journey and could not have been accomplished without the generous help and support of many people. I would like to especially thank the following people: •  A l a n K r a u s , m y thesis supervisor, for his guidance and unconditional support. H i s encouragement and advise helped me from the beginning to the end of this research. H e assisted me i n the understanding of the problems and provided me w i t h insightful suggestions of how these problems should be approached.  •  R o n Giarnmarino, for providing me w i t h intellectual and academic insights that helped me focus on the important issues.  •  For A m i r Barnea, Gilles Chemla, A d l a i Fisher, Lorenzo Garlappi, R o b Heinkel, T o m Ross, M a r k Rubinstein, Jacob Saggi, Issouf Soumare, and R a l p h W i n t e r for valuable suggestions and advice.  vni  CHAPTER I WHAT DO SHAREHOLDERS REALLY WANT? INCENTIVE CONTRACTS A N D DEBT POLICY W H E N INVESTORS A R E DIVERSIFIED 1.1  Introduction One of the major achievements of modern investment theory is our under-  standing of the importance of diversification. T h i s theory, initially pioneered by M a r k o w i t z (1952) and Sharpe (1964), explains why it is common practice for most investors to hold diversified portfolios. G i v e n that portfolio diversification is a basic concept i n finance and given the widespread empirical phenomenon of diversification , it is surprising that most financial economists still consider that 1  the goal of the firm is to maximize its own value. In broad terms, the corporate finance literature tends to neglect the fact that if companies affect each other's cashflows, the goal of diversified investors is not firm value maximization, but rather m a x i m i z a t i o n of the value of their portfolio. T h e fact that managers are typically not diversified and are compensated according to the performance of the firm they manage results i n an agency problem which is the focus of this chapter.  W e argue that managers who maximize firm value are acting subop-  t i m a l on behalf of their diversified investors whenever firm value m a x i m i z a t i o n implies engaging i n actions that reduce the value of other firms i n the investor's portfolio. In this chapter, we show how shareholder's diversification affects the choice of 2  managerial compensation and debt policy when a company's cashflow is affected I n 1989, institutional investors (pension funds, mutual funds, insurance companies, bank trust, and foundation/endowments funds) had a combined holding of 53.2% of corporate America equity (Brancato, 1991). It is hard to know what is the exact percentage holding today, but there is evidence that the percentage is much higher (Porteba, 1998). Hansen and Lott (1996) present evidence that institutional investors are highly diversified. For our purposes, it is also important to understand that most institutional investors hold shares in firms that affect each other (i.e., i n the same industry). For example, 77% of Intel and 71% of Compaq are owned by institutions that have holdings i n at least one of the other five big computer industry companies; 56% percent of Chrysler is held by institution that simultaneously hold shares i n Ford and/or G M ; 34% of the combined market value of Ford, G M , and Chrysler is held by institutions holding all 3 firms; many mutual funds invest i n a certain industry, which by definition means that they invest i n competing firms. 1  Throughout the paper, we assume that shareholders are homogeneous, perfectly diversified, and hold an equity stake in the market portfolio. This is done for simplicity and tractability and suits our objective, which is to counter the typical corporate objective that assumes undiversified shareholders. It is also consistent with asset pricing models such as the CAPM. However, having heterogenous investors with different portfolio holdings does not matter for the qualitative nature of our results. The only common objective that unites all diversified investors is the maximization of the combined value of firms in the economy, or stated i n other words, the growth in the economy. O n the other hand, maximization of firm value is not a common 2  1  by other companies' actions. In our set-up, a non-diversified, risk-neutral manager affects the cashflow of the firm. T h e manager's effort choice is a two dimensional problem. T h e manager exerts cannibalistic effort and economy-increasing effort. T h e cannibalistic effort involves imposing a negative externality on other firms. T h i s would typically involve taking market share from a competing firm, but could also be any negative externality that affects firms from w i t h i n the industry or outside the industry (e.g., legal disputes w i t h suppliers, competition i n the labor market, etc.). Contrary to this, economy-increasing effort involves investing i n new markets that enhance the opportunity set and does not affect other firms i n the market. of differences.  3  W e assume that the two effort types exhibit two kinds  F i r s t , cannibalistic effort increases the cashflow of the firm by  reducing the cashflow of other firms i n the economy, while economy-increasing effort increases the cashflow of the firm without affecting the cashflow of other firms i n the economy. Second, we assume that economy-increasing effort is riskier t h a n cannibalistic effort. T h i s reflects the idea that economy-increasing effort is exerted i n new markets w i t h very little information about future prospects, while cannibalistic effort is exerted i n mature markets w i t h more predictability. Interpreting to Bayesian terminology, when a firm engages i n cannibalistic projects, the estimates of cashflow depend on past observations and therefore are rather precise. Contrary to this, when the firm engages i n economy-increasing projects, there are no past observations to learn from, resulting i n imprecise estimates of cashflow, which exhibit a high degree of variance.  4  O u r theoretical analysis concerns different incentive contracts and their effect on shareholder's portfolio value. W e show that when the manager is compensated according to firm value w i t h equity she tends to exert cannibalistic effort even though shareholders prefer that the manager would engage only i n economyobjective. There is no reason to believe that the diversified shareholders of a specific firm care more about its value than the value of the firm's competitor. Note that both type of manager's effort affect welfare. Economy increasing effort results in the development of new markets and products, while cannibalistic effort increases welfare by competition within the same market, i.e., price cuts, small improvement in product quality, etc. Therefore, not dealing with the total welfare aspect does not qualitatively alter our results. In fact, our model builds on the idea that real investment decisions are a form of competition. A firm can generate new cashflow either at the expense of other firms, or by innovation and development of new products and markets. Diversified shareholders prefer the later. 3  The pharmaceutical industry presents a rather pure example of cannibalistic vs. economy increasing. In general, pharmaceutical companies can take the riskier track of trying to enhance the opportunity set by finding a cure to an untreated disease or go on the safer cannibalistic track and create their own generic drug for a disease that is already treatable. (Pharmaceutical Executive, September 2002). In general, most real investment opportunities combine cannibalistic and economy increasing aspects in different proportions. For convenience, we assume that managers can engage in any desired mix of these aspects. 4  2  increasing effort.  5  Cannibalistic effort does not create value for shareholders  because they hold shares i n the negatively affected firms as well. W e analyze how debt may reduce this conflict. W e show that the possibility of bankruptcy results i n a relatively greater loss i n payoff to cannibalistic effort t h a n i n payoff to economy-increasing effort.  T h i s helps achieve the objective  of shareholders to shift the manager's choice of effort towards relatively more economy-increasing effort.  W h i l e Jensen and Meckling (1976) show that risk-  shifting is a form of agency cost, i n our model risk-shifting helps induce managers to engage i n economy-increasing projects.  Thus, debt plays a positive role i n  mitigating the agency conflict. W h i l e others have showed that debt can induce monitoring activity of managers and reduce the free cashflow for managers to engage i n perk consumption (e.g., Jensen 1986), we show that debt can create value for the very same reason that Jensen and M e c k l i n g (1976) showed that it reduces value. In other words, risk-shifting creates value because it induces the manager to engage i n projects that increase portfolio value. O n the other hand, debt incorporates two costs. F i r s t , since shareholders decide b o t h on the capital structure and the managerial compensation policy, they still have the incentive to transfer wealth from debtholders to themselves. Thus, the inability to commit to a certain incentive policy results i n a conflict between shareholders and debtholders. Since debt is priced i n the market by debtholders, shareholders pay a price for this commitment problem. Second, similar to Myers' (1977) debt overhang, debt incorporates a cost of under-investment. T h e overall effect of debt is that shareholder portfolio value may either increase or decrease. W e also analyze the possibility of compensating the manager by stock option grants. W e show that option grants have a l l the advantages of debt but do not expose the firm to the conflict between shareholders and debtholders. Moreover, we show that option grant incentives are better t h a n equity compensation i n creating value for shareholders. T h i s is because option grants are a better t o o l t h a n equity i n promoting the high variance economy-increasing projects. W e stress the normative value of our results. Just as investors were not aware of the importance of diversification prior to M a r k o w i t z (1952), there is no reason The literature on agency problems between managers and shareholders takes two main views (Grinblatt and Titman, 2001): (1) managers take advantage of their positions and engage in actions that allow them to benefit personally at the expense of shareholders (perk consumption). (2) Managers view their positions as serving a broader constituency than just shareholder, i.e., employees, customers, and other stakeholders. Note that our framework has the properties of both types of agency conflicts. While cannibalistic effort is beneficial to managers simply because they are not diversified and therefore their wealth is affected by it (similar to perks), it is also beneficial to workers who care mostly about the firm's ability to pay wages and stay solvent. 5  3  to believe that boards of directors and policy makers are aware of how shareholders' diversification should affect monitoring activities and policy making. O n the contrary, we believe that boards of directors perceive their job as to monitor management and make sure that the C E O pursues projects that maximize firm value. O u r implications are therefore better suited to provide guidelines for public and corporate policy. E v e n so, we point to some empirical evidence that is consistent w i t h our theory. For example; (a) the simultaneous tendency i n the last two decades of shareholders being more diversified (Saxton, 2000) together w i t h management being compensated more by stock option grants (Hall and L i e b m a n , 1998, and M u r p h y ,1999) is consistent w i t h the fact that compensating managers by options tends to increase portfolio value for diversified shareholders; (b) since growth firms tend to have more economy-increasing projects, our model implies that C E O ' s of growth companies should be compensated more closely to their companies' performance. C l i c h (1991), S m i t h and Watts (1992), and Gaver and Gaver (1993) found that equity and options are used more extensively i n the compensation of executives i n growth  firms.  6  O u r approach is related to several strands of literature.  In the traditional  agency literature (Hart and Holmstrom, 1987, B o l t o n and Dewatripont, 2002), a principal, say a shareholder, who is concerned about firm value m a x i m i z a t i o n is faced w i t h a trade-off between motivating a risk-averse manager to increase firm value and the cost of having the manager bear risk. W h i l e the traditional agency problem is solved i f the manager is risk-neutral by compensating the manager based only on firm value and letting the manager bear a l l the risk, our agency problem exists even i f the manager is risk-neutral. T h i s is because also a risk-neutral manager engages i n cannibalistic projects when he is undiversified. Therefore, to abstract from the traditional agency problem, we present and analyze our agency conflict i n a set-up where the manager is risk-neutral. Note also that a major aspect of the executive compensation literature deals w i t h the nonsystematic risk that managers are exposed to due to incentives (e.g., Meulbroek, 2001); however, this literature is silent on the problem that arises because the real objective of shareholders is not firm value m a x i m i z a t i o n but portfolio value maximization. W e recognize two important aspects of financial markets: (a) F i r m s interact i n imperfectly competitive markets and affect each other, (b) the shareholder's objective is to maximize the value of a diversified portfolio. W h i l e , there are a few papers that show how capital structure and compensation choices affect product Bizjak, Brickley, and Coles (1993) and Gaver and Gaver (1995) found no significant relation between growth opportunities and compensation in their samples. 6  4  market interaction (Aggrawal and Samwick, 1999, B o l t o n and Scharfstein, 1990, Brander and Lewis, 1986, Maksimovic, 1988, Rotemberg and Scharfstein, 1990), none of t h e m considers the fact that shareholders are diversified. In some respects the conflict of interest that we analyze is a direct application of the literature on shareholder unanimity (e.g., L o n g , 1972, E k e r n and W i l s o n , 1974, DeAngelo, 1981, Hirshleifer and Riley, 1992). T h e conclusion of this literature is that shareholders' unanimity i n support of firm value m a x i m i z a t i o n can only be achieved i n a perfectly competitive environment i f a project's cashflow can be spanned by existing securities i n the capital markets (Jensen and L o n g , 1972, M e r t o n and Subrahmanyam, 1974). E v e n though there are disputes on the unanimity conclusion (Grossman and Stiglitz, 1977, Rubinstein, 1978), for our purposes, it is only important to realize that the literature i n industrial organization and labor economics makes it clear that product and labor markets are not perfectly competitive (e.g., Tirole, 1988, C a r l i n and Soskice, 1990). U n d e r such circumstances the unanimity conclusion fails, the NPV  rule disregards the  cashflow generated tp other firms i n the economy, and shareholders are concerned w i t h portfolio value which is different then firm value.  7  Another set of related papers are theoretical studies that either show why a convex payoff contract such as executive stock options may have a positive role i n financial markets (Haugen and Senbet, 1981, S m i t h and Stulz, 1985, Farmer and W i n t e r , 1986), or those that show that i n a dynamic framework it is not clear whether options increase managerial risk appetite (Detemple and Sundaresan, 1999, Carpenter, 2000). O u r chapter takes a different approach, instead of showing why options do not increase risk taking behavior of managers, we show that contrary to Jensen and Meckling (1976) increased managerial risk taking creates value for diversivied shareholders. Under such circumstances, executive stock options have a natural role i n financial markets. Finally, there are a few studies that deal w i t h portfolio diversification and how it affects firm value ( A d m a t i et al., 1994, D e M a r z o and Urosevic, 2000). These studies assume that a large shareholder who maximizes his own portfolio monitors the firm and affects its value. However, they do not address the issue of portfolio maximization versus firm value maximization. T o the best of our knowledge, only Hansen and L o t t (1996) address explicitly the fact that i f companies impose externalities on each other, then diversified shareholders do not want to maximize firm value. However, they do not address the agency problem, and they do not relate the concept of portfolio m a x i m i z a t i o n to issues of incentives and capital Moreover, if we assume perfectly competitive markets, the NPV of any project is zero. Under such conditions, a firm's real investment decisions are irrelevant. 7  5  structure, which is our focus. T h e remainder of the chapter proceeds as follows. In Section 1.2 we develop the basic model, where a risk-neutral manager is given stock incentive to exert the two types of effort i n an all equity firm. In Section 1.3 we extend our analysis to the case of a leveraged firm and analyze the effect of debt on the effort choice and shareholder's portfolio value. W e compare the all equity and leveraged firm and show under which circumstances debt can increase value for shareholders. I n Section 1.4 we study the case of stock option grants to managers and show why they are the o p t i m a l contract. Section 1.5 is a discussion on the empirical implications of the model. Finally, Section 1.6 concludes. T h e appendix of the chapter contains all proofs.  1.2  A l l Equity Firm W h e n shareholders are perfectly diversified, they care about the total value of  the market. Thus, i f there was no incentive problem (i.e., the regular principalagent problem), compensating managers according to the market portfolio would have been a good idea. Some would argue that an incentive scheme that relies partly on other firms i n the industry might be a plausible solution. For example, suppose that Pepsi and Coke are engaged i n a costly aggressive advertising scheme to change consumer tastes towards their company's product.  A com-  pensation package that takes into account the combined value of the two firms may be preferable for shareholders.  However, for all practical purposes, such a  compensation package results i n Pepsi and Coke becoming one firm and we believe that anti-trust laws and agencies would reject it. O u r analysis, therefore, concerns incentive contracts that are related only to the specific firm value.  8  W e consider firm A w i t h cash-in-place of w, which is also the managerial competitive wage. A t t = 0, shareholders set a fixed wage w  and an o p t i m a l  share of the firm a > 0 as compensation for the manager.  T h e cash of the  0  firm can be used to pay some or all of the manager's fixed wage, while the rest is distributed untaxed to shareholders. manager of firm  A t t = 1, given a, the risk-neutral  A decides on the two types of effort: cannibalistic effort, E , and c  economy-increasing effort, E .  9  s  T h e manager can exert effort Ei G { 0 , c o } , i G  Aggarwal and Samwick (1999) derive managerial contracts that are dependent positively on the performance of the industry and facilitate collusive behavior. They argue that firms that hold passive equity positions in rival firms do not violate antitrust laws. Note however that even if compensation based on a competing firm is legal, it may still destroy value for shareholders if the incentive of firms to pursue new projects is reduced (e.g., Arrow 1962). 8  9  The timing is not important in our setting. We assume it only for convenience. In general all we need 6  {s,c}  at cost \Ef.  T h e two-dimensional effort choice gives the manager control  over project selection. T h e manger can either exert cannibalistic effort that hurts other firms, or invest i n economy-increasing effort that does not affect other firms. H o l m s t r o m and M i l r g r o m (1987) emphasize that a key restriction of the standard agency model is that the manager has no discretion over project selection, or, equivalently, that the shareholders specify which project the manager chooses. Addressing this problem, i n our model the manager has two types of projects and decides on the m i x of projects. A t t — 2, the cashflows of the firms i n the economy are generated and the payoffs are distributed to the manager and the shareholders. T h e two types of effort affect b o t h the expected value and risk properties of firm A ' s final cashflow. M o r e specifically, the cashflow generated by cannibalistic effort is cE , while economy-increasing effort payoff follows a binomial distribuc  t i o n such that the payoff may be either s E h  s  i n the high state or siE  s  i n the low  state w i t h probability (1 — 6) and b respectively. Thus, the expected cashflow and variance of the firm is [i  A  a\  =  (s (l -b)  =  sE  =  b(l-b)(s - ) E>  + s b) E + cE  h  s  t  s  c  + cE  (1.1)  c 2  h  Sl  (1.2)  where s is the expected cashflow generated by a unit of economy-increasing effort and c is the cashflow generated by a unit of cannibalistic effort. Note that only the economy-increasing effort E  s  affects variance of cashflow, while cannibalistic ef-  fort is completely deterministic. T h i s reflects the idea that economy-increasing effort is exerted i n new markets w i t h very little information about future prospects, while cannibalistic effort is exerted i n known markets w i t h more predictability. T h e link between less information and a more uncertain cashflow can easily be established by Bayesian methodology. Consider a firm concerned w i t h obtaining accurate estimates of the cashflow expected from economy-increasing and cannibalistic effort. T h e firm is rather certain, a priori, about the cashflow generated from cannibalistic effort because it can observe past data. O n the other hand, there are no observations of cashflow i n a market that has not yet been formed. Thus, estimates of cashflow from cannibalistic effort are more accurate compared to estimates of cashflow generated from economy-increasing effort. In the limit, cannibalistic projects cashflow of a mature market are completely deterministic, while cashflow of economy-increasing projects are uncertain because there are no is the standard moral hazard assumption that manager's effort choice is unobservable. 7  prior observations (Box and Tiao, 1973). In our framework, we assume that the risk a\ is non-systematic and therefore completely diversifiable.  10  T h e manager choice of effort does not only affect firm A's cashflow, but also affects the cashflow of other firms i n the economy. A n increase i n firm A's cashflow due to cannibalistic effort comes at the expense of a reduction i n cashflow of other firms i n the economy by cE . c  Thus, the cashflow of the other firms i n the economy  is the cashflow that exists without the existence of firm A, minus the cashflow that is taken by firm A due to its manager's cannibalistic effort.  Note that  cannibalistic effort is a negative externality imposed by firm A on other firms i n the market, while economy-increasing effort does not impose any externality on other firms i n the m a r k e t .  11  We solve the dynamic model by backward induction. A t t — 1, the manager of firm A, chooses the two types of efforts, so that they maximize her utility UM-  U  M  = max w + a\i  - -(E  + E)  (1.3)  2  2  A  0  C  S  Solving for the optimal effort choice we get,  E  c  =  ac  (1.4)  E  s  =  as  (1.5)  To avoid cases of a negative effort, we assume that c,s > 0. Note that as long as c > 0, the manager exerts cannibalistic effort.  Since the manager is not  diversified, she is not affected by the negative externalities that cannibalistic, effort imposes on the other firms i n the economy. 1 0  12  W e let 9 be the conflict e  Note that there is no reason to believe that this risk is not diversifiable.  A more structured model would allow for positive externalities (i.e., E having a positive effect on other firms). This would only make the mathematical representation more complex, but would not change the qualitative nature of our results. The important thing is that shareholders benefit only from economy increasing effort. 1 1  s  There are three more reasons which make the manager favor cannibalistic projects to a larger extent than is captured by the model: (1) If the manager is risk-averse, the safer cannibalistic projects are preferred. (2) the fact that a manager desires to remain on the job provides a reason for taking on the safe projects in the mature market. If the manager takes on the risky project there is a possibility that a new market will be formed. Thus, there are chances that new expertise will be needed, and the manager may become expendable in the future. Grinblatt and Titman (2001) argue that for this reason, oilfirmsmay have continued to invest in oil exploration in the early 1980s despite falling oil prices. (3) In general, risky projects may drive firms to bankruptcy. Under such circumstances, the manager tends to suffer a high personal cost. Gilson (1990) reported that only 43 percent of the chief executive officers keep their jobs subsequent to the bankruptcy of their firms 12  8  measure i n an a l l equity firm.  E  (1.6)  s  s  T h e conflict measure quantifies the degree of the conflict of interest between the manager and the shareholders. T h e higher is 9 , the more severe is the conflict e  of interest. W h e n the manager engages i n cannibalistic effort shareholders  are  not better off because the cashflow of firm A increases at the expense of other firms i n the economy. O n l y when c = 0 there is no conflict of interest because the manager does not engage i n cannibalistic effort at a l l . A t t = 0, shareholders choose the incentive a subject to the manager's getting her reservation wage. T h e o p t i m a l incentive a is that which maximizes firm A's contribution to shareholder's portfolio value.  Thus, shareholders are not con-  cerned w i t h the value of firm A, but rather they maximize firm A's marginal contribution to the value of their portfolio. We represent the marginal contribut i o n of the a l l equity firm by V , e  V  e  =  m a x (—cE ) + (1 — a)fi c  A  + (w — w ) 0  (1.7)  a  S.t.  U,m  >  W  T h e first t e r m represents the reduced cashflow of the other firms i n the economy due to the cannibalistic projects of firm A, while the second and t h i r d terms are the residual cashflow for the shareholders of firm A after payment to manager. E v e n though shareholders are risk-averse, since we assume the economyincreasing risk to be non-systematic it does not affect the shareholder's terminal wealth. It is important to realize that shareholders are not indifferent to cannibalistic effort.  T h e y suffer a deadweight cost of acE because part of firm c  A's cashflow belongs to the manager. Substituting for the o p t i m a l effort choices E ,E , C  S  we are able to derive the o p t i m a l incentive a and firm A ' s contribution  to shareholder's portfolio value V . e  P r o p o s i t i o n 1 In equilibrium, the optimal incentive a and firm contribution to shareholder's portfolio V are e  a  2 (c + s ) 2  2  3s  (1.8)  4  8 (c 9  2  + s ) 2  (1.9)  Both the incentive contract a and firm A's contribution to shareholder's value V increase with the expected payoff from economy-increasing e  portfolio  effort s and  decrease with the payoff from cannibalistic effort c. T h e intuition behind proposition 1 is straightforward. Shareholders want the manager to engage only i n economy-increasing effort E . T h e y would prefer that s  the manager would not invest any cannibalistic effort at a l l because it reduces their portfolio value. T h e higher the expected payoff from economy-increasing effort and the lower the payoff from cannibalistic effort, the lesser is the conflict of interest between the manager and the shareholders and the lower is the conflict measure 9 . T h e result is that a and V increase w i t h s and decrease w i t h c. Note e  e  that i n the extreme case, when c = 0, there is no conflict of interest and the model is reduced to the case where firms generate independent cashflows and do not affect each other. Under such circumstances maximization of firm value is the same as maximizing shareholder's portfolio value and | j j = ^ ^ ^ = 0. 2  2  Thus, i n this reduced framework, the optimal incentive scheme is constant and does not depend on the expected cashflow s.  1.3  A Leveraged F i r m  Debt and incentives F i n a n c i a l markets have evolved i n recent years i n ways that lessen the more significant problems. In most companies, for example, the debt level choice is a decision made at the board of directors' level. Hence, firms w i t h active outside board members can force managers to select a debt level higher t h a n that which the managers would personally prefer. W e analyze how debt-in-place may affect the conflict between the manager and shareholders. A t t = 0, the firm issues a face value D of risky debt, where cE + siE c  s  < D < cE + ShE . c  s  T h e debt is priced i n the competitive  financial  markets and the proceeds of the debt issue are distributed to shareholders as dividends. W e assume a world w i t h no taxes, so dividends are not taxed. A t t = 1, shareholders choose the fixed wage w  0  and the incentive share ad that  they give to the manager. A t t = 2, the manager of firm A, decides on the two types of effort. A t t — 3, the cashflow of the firm is generated and the payoffs are distributed to the manager, shareholders, and debtholders. T h e t i m i n g of the model is illustrated i n Figure 1.1, and the t = 3 cashflows to shareholders, debtholders, and manager i n the two possible states (bankruptcy and solvency) is illustrated i n Figure 1.2. W e impose three restrictions on the model's parameters: 10  A s s u m p t i o n 1: s\ > c + 2  2siSh-  T h i s condition makes the model w i t h debt-  in-place viable. If the condition is not met, the manager participation constraint is not met when there is debt-in-place. Assumption  2: ^  > ^ / ^ j y . T h i s condition is required i n order that the  o p t i m a l debt face value be positive. 2  A s s u m p t i o n 3: si >  T h i s condition is required i n order that the low  state cashflow be positive, i.e., cE + s E c  t  > 0. In other words, it should be clear  s  that the manager, shareholders, and debtholders are not subject to liabilities at the end of the game. Since s and c are non-negative, the only parameter that can be negative is si but up to this m i n i m u m bound. A s w i t h the a l l equity firm, we solve the model by backward induction. A t t = 2, the manager of firm A chooses the two types of efforts, so as to maximize her utility. Note that since there is debt-in-place the expected cashflow for the current owners of firm A (i.e., shareholders and the manager) is H = (l-b)(s E A  h  + cE -D)  a  (1.10)  c  T h i s reflects the fact that w i t h probability b, the firm w i l l be bankrupt. U n d e r such circumstances b o t h the manager and shareholders receive nothing, and the claimants are the debtholders. T h e manager maximizes her utility and chooses the two efforts U  M  =  m a x w + a (l 0  - b)(s E  d  h  + cE - D) -\(E  2  s  c  + E ) 2  C  S  Solving for the o p t i m a l effort choice we have, E  c  =  a c(l-b)  (1.12)  E  s  =  cxMl-b)  (1.13)  d  In broad terms, allowing for a bankruptcy state has the negative effect of reduced endeavor , which is captured by the factor (1 — b) i n eq. (1.13) . Note that from 13  the point of view of the shareholders, only a reduction i n economy-increasing effort is a cost because a reduction i n cannibalistic effort is value creating. Therefore, the reduced endeavor cost refers only to the possible reduction i n economyincreasing effort. T h i s problem is closely related to Myers (1977) debt overhang (under-investment).  Since the manager knows that under bankruptcy she re-  ceives nothing for her efforts, she tends to exert less of i t .  1 4  T h e higher is the  Note that in Myers (1977) projects are not taken on because no one is willing to finance them. Somewhat similarly, in our framework, the manager exerts less effort because under bankruptcy she receives no payoff for her effort. 14  11  probability of bankruptcy, the less effort she tends to exert. In an a l l equity firm there is no risk of bankruptcy so (1 — b) = 1. In the leveraged firm, the reduced endeavor cost is reflected i n the (1 — 6) factor being less t h a n 1. W e refer to the conflict measure of a leveraged firm as 6d and it equals  (  U  4  )  C o m p a r i n g to eq. (1.6), we observe that 0 < 9 . Thus, the conflict of interest d  e  between the manager and shareholders is reduced due to the issuing of debt. T h e conflict measure decreases because the relative loss i n cannibalistic effort payoff is greater t h a n the relative loss i n economy-increasing effort. T h i s is because i n the low state cannibalistic effort payoff is the same as i n the high state and equals cE , while economy-increasing effort payoff is lower i n the low state relative to c  the high state, i.e.,  siE < ShE . T h e fact that the manager cares only about s  s  the outcome of the high (solvency) state makes economy-increasing effort more valuable to her compared to cannibalistic effort. Basically, our model shows that risk-shifting may create value for shareholders. W h i l e the literature starting w i t h Jensen and Meckling (1976) regards riskshifting as a form of agency cost between shareholders and debtholders, we show that risk-shifting serves as a way to mitigate the conflict of interest between the manager and the shareholders. Jensen and Meckling (1976) wrote: "with debt  the owner-manager w i l l have a strong incentive to engage i n  activities (investments) which promise very high payoffs i f successful even i f they have a very low probability of success. If they t u r n out well, he captures most of the gains, if they t u r n out badly, the creditors bear most of the costs." W h i l e Jensen and Meckling (1976) recognized the incentive of the ownermanager to engage i n riskier investments once debt is i n place, they d i d not consider that this may create value for shareholders. In broad terms, the ability to risk shift creates value by switching the managerial choice of projects to those that increase portfolio value for shareholders.  W h e n shareholders are diversi-  fied, the agency cost of Jensen and Meckling (1976) is only partially a problem. Jensen and M e c k l i n g (1976) recognized shareholders' incentive to transfer wealth from debtholders to themselves. However, the idea that this results i n t a k i n g on negative net present value (hence, NPV)  projects, which distorts value is wrong.  T h i s is because shareholders are not concerned w i t h the expected cashflow of firm A only, but rather are concerned w i t h the total cashflow generated i n the economy. T h e y take on debt to align the manager w i t h that objective and understand that this may reduce the expected cashflow of firm A. W e believe that this objective is consistent w i t h shareholders being diversified, while the 12  NPV  calculation (as typically applied) is not. T o understand this point, note that the NPV of a project is calculated by estimating the expected cashflow for a specific firm a n d discounting it by the cost of capital. W h i l e the cost of capital is calculated under the assumption that shareholders are diversified, the expected cashflow is only that of the firm. Under such circumstances, taking o n negative NPV  projects does not distort value for shareholders because the calculation  ignores the cashflow generated to other firms i n the economy. A t t = 1, shareholders choose an o p t i m a l incentive a  15  that maximizes the  d  contribution of leveraged firm A to their terminal portfolio value. Vid =  m a x (-cE ) + (1 - a )n c  d  + (w - w )  A  (1.15)  0  Old  =  max(-cE ) c  + (l-a )(l-b)(s E d  h  + cE -  s  D)  c  ad-  Substituting for the o p t i m a l effort choices E ,E , C  m a x -a (l-b)bc -a (l-bfc 2  2  we have  S  + a (l -a )(l-bfs  2  - (1 -cx )(l-b)D  2  d  d  d  (1.16)  d  T h e first order condition can be written  § ^ oa  = - ( 1 - b)bc - 2a (l - bfc + (1 - 2a )(l - b) s + (1 - b)D = 0 (1.17) 2  2  2  d  d  2  h  d  which gives the incentive a i ^ .  16  s (l-b)-bc 2  a  d  D  2  2(1 - b) (a + c ) 2  T h e manager's incentive share a  2  d  +  2(1 - b) (s +c ) 2  2  {  -  °  j  increases w i t h the debt face value D. T h e  fact that the manager's wage per share is reduced due to debt, enables shareholders to increase managerial effort choice by increasing the incentive share a . d  A t t = 0, the debt is valued i n the competitive debt market at the price of p . d  p = {ld  b)D + b (cE + siE ) c  (1.19)  a  Note that the NPV calculation ignores any agency problems and measures the contribution of a project to shareholder's wealth. However, perhaps the best way to understand the agency problem of this study is by noting that the numerator of the NPV calculation (as typically applied at the firm level) is the expected cashflow of the firm, which would be advanced by undiversified managers but not diversified investors who care about the value of their portfolio, while the denominator is the discount rate demanded by diversified investors but not that demanded by undiversified managers. 1 5  1 6  Clearly, the first-order condition defines a unique maximum, as g^l < 0. d  13  d  Debtholders price the debt based o n the ability of the firm to return its obligation. Thus, they analyze the expected cashflow of their claim and do not care whether the cashflow comes from the firm engaging i n cannibalistic effort or economyincreasing effort. Engagement i n more economy-increasing effort increases the riskiness of the debt and that is reflected i n the price of the debt issue. Shareholders choose the optimal debt level subject to the manager's participation constraint,  V  d  =  max (p + Vid) + (w - w ) d  (1.20)  0  s.t. UM  >W.  Substituting for the optimal effort choices  E ,E , we derive the o p t i m a l C  S  debt  face value D, the incentive a , and leveraged firm A's contribution to shareholders d  portfolio V . d  Proposition 2  The optimal debt face value D and incentive a  d  4(1  are  -b)-bc  2  3 V(l-6)(4 + c ) 2  which results in firm's contribution 2 (4(1  to shareholders being  - b) - be ) (24(1 2  - b) + be + 3bs,s ) 2  h  The face value of debt D increases with Sh, decreases with c, and is unaffected by s/. The incentive a  increases with Sh, decreases with c, and is unaffected by si.  d  The firm contribution  to shareholders portfolio V increases with s^ and si, and d  decreases with c. A n increase i n s  h  reduces the conflict of interest between the managers and  the shareholders. T h i s can be seen b y the lower conflict measure 6 a n d results d  i n comparatively more economy-increasing effort. conflict, shareholders set a higher incentive a  d  A s a result to the reduced  and a higher debt face value D.  T h e higher debt face value D allows shareholders to increase managerial choice of effort through the incentive a without actually increasing the manager's wage d  above the reservation wage.  17  C o n t r a r y to the increase i n Sh, a n increase i n c  Note that issuing debt results in debtholders being entitled to some of the firm's cashflow, so the objective of the "average" claimant changes. Debtholders, similar to the manager, value cannibalistic projects because it affects the firm's ability to pay its claim. 17  14  increases b o t h the post debt issue conflict between the manager and shareholders (seen by the higher 6d) and the post debt issue conflict between shareholders and debtholders. Shareholders reduce the incentive a  d  and reduce the debt face value  D. T h e comparative static results show that debt face value D and incentive ad are not affected by s . T h i s is because shareholders cannot commit to their t  incentive share ad which is set after debt is i n place. Thus, once debt is i n place, shareholders have an incentive to set managerial compensation differently t h a n what would be desired by debtholders, who are the sole recipients of the cashflow under bankruptcy. T h e result is that when setting the incentive o^, shareholders disregard the low state payoff and it does not depend on si. T h i s means that the debt face value D does not depend on si because it depends on the incentive ad and chosen to satisfy the manager's participation constraint. O n l y the pricing of the debt pd is affected by si, which i n t u r n affects the firm's contribution VdT h e rest of the comparative static results are straightforward. T h e firm cont r i b u t i o n to shareholder's portfolio Vd increases w i t h Sh and s , and decreases t  w i t h c. T h i s is because only economy-increasing payoff adds value to portfolio value, while cannibalistic effort decreases portfolio value. A l l equity compared to a leveraged firm W h i l e debt reduces the conflict between the shareholders and the manager, it is also subject to two forms of cost. F i r s t , the fact that the manager does not earn any wage i n the low state (bankruptcy state) results i n the reduced endeavor cost. Second, debtholders know that the incentive share ad given to the manager at t = 1 by the shareholders, w i l l promote the objective of shareholders who at the time ignore the low state payoff. T h i s commitment cost w i l l be reflected i n the debt price at t = 0. Thus, shareholders may not be able to use debt as a t o o l for creating portfolio value i f these two costs are high compared to the benefits of the reduced conflict between the manager and shareholders. To understand these forces we analyze Figure 1.3. T h e figure plots the cont r i b u t i o n value of a leveraged firm compared to an a l l equity firm for different levels of payoffs. T h e numeraire is Sh = 1, while the x-axis quantifies the level of cannibalistic effort payoff c, and the different curves represent the level of economy-increasing effort payoff i n the low state si. To understand the trade off we start from the simple case where c = si = 0. Under such circumstances, the issuance of debt does not carry on any reduced endeavor cost.  T h i s is because whether there is debt-in-place or not has no  affect on the low state payoff, which is always zero. E v e n i f the firm is a l l equity 15  financed, the manager essentially disregards the low state payoff when deciding on the effort level. Note also that when c = si = 0, the objective of shareholders i n b o t h an a l l equity firm and a leveraged firm is to maximize the expected cashflow of the firm. W h e n c = 0 the firm does not affect the cashflow of other firms i n the economy, while c = si = 0 means that disregarding the low state payoff does not affect the expected cashflow of the firm. There is no conflict between debtholders and shareholders because the low state produces a zero payoff i n b o t h an a l l equity and leveraged firms. T h e result is that debt financing has only the positive effect of increasing effort level. B y issuing debt and setting a higher incentive a  d  t h a n is possible i n an a l l equity firm, shareholders are able  to increase the cashflow of the firm. W e can see that the increase i n contribution value for a leveraged firm compared to an a l l equity firm is almost 20%. W h e n c = 0, but si > 0 there is a cost for having debt-in-place.  Since  the manager disregards the low state payoff, the manager tends to exert less economy-increasing effort which is reflected i n the reduced endeavor cost. T h i s is especially important i f the probability b of the low state occurring is high. Note that si does not affect the incentive a . T h i s is because shareholders do not d  receive the low state payoff. T h e result is that the incentive a  d  i n the leveraged  firm is always higher t h a n the incentive a i n an a l l equity firm. Thus, there is a trade-off between the ability of having debt-in-place and increasing incentive and to cost of the reduced endeavor. T h e cost of the reduced endeavor becomes more severe as si increases . T h e analysis becomes more involved when one considers the more interesting case when the firm can affect the cashflow of other firms, i.e., c > 0. W e start by assuming that si = 0. A s c increases, the benefits of having debt-in-place become more apparent because the conflict between the manager and shareholders is reduced w i t h debt, i.e., 9  d  < 9 . B y shifting the manager choice of e  effort w i t h debt, shareholders induce the manager to take on comparatively more economy-increasing projects, which increase portfolio value, and comparatively less cannibalistic projects, which reduce portfolio value. Moreover, the reduced endeavor cost is zero. Since si = 0, the fact that the manager disregards the low state payoff does not reduce the amount of economy-increasing effort. T h e only cost that exists is the commitment cost. Debtholders price the debt according to their rational expectation about the cashflow of the firm. T h e y correctly assume that at t = 1 the incentive is set i n a way that produces the highest value for shareholders. U p to a certain level of c, a leveraged firm creates value compared to an a l l equity firm. T h i s is because the benefits from the shift towards economy-increasing projects outweigh the costs of the commitment problem w i t h 16  debtholders. It is important to understand that the fact that the manager engages i n riskier projects and that debt is valued accordingly is not i n itself a problem. T h e cost of having debt-in-place comes from the commitment problem. Shareholders set the incentive a  d  i n a way that maximizes their own portfolio  value, even though it may reduce the expected cashflow of the firm. A s c increases, the incentive a  d  is reduced. T h i s is because portfolio considerations make share-  holders worse off when cannibalistic effort payoff is high. E v e n though this is also true i n an a l l equity firm, the deadweight cost due to cannibalistic effort is higher i n a leveraged firm.  In an a l l equity firm cannibalistic effort results i n  a deadweight cost of a times the cashflow generated by cannibalistic projects. T h i s is the portion of the firm's cashflow that belongs to the manager.  In a  leveraged firm the shareholders get a (1 — a ) share of the cashflow only i n the d  high state, while the debtholders are those who receive a l l the cashflow i n the low state. Thus, i f bankruptcy occurs, a l l the cashflow which is generated from cannibalistic projects is a deadweight cost from the point of view of shareholders at t = 1. Thus, as c increases, shareholders set a lower incentive a . However, d  a lower incentive a  d  does not only reduce cannibalistic effort, but also reduces  economy-increasing effort.  T h e result is that the firm generates a low level of  cashflows. Debtholders anticipate this outcome and price the debt accordingly. Thus, at some level of c the inability of shareholders to commit to their incentive scheme, results i n debt reducing the contribution value compared to an a l l equity firm. W h e n b o t h c > 0 and si > 0 the benefits from the reduced conflict and the cost of reduced endeavor and commitment a l l play a role. In broad terms an increase i n c results i n b o t h benefits and costs for a leveraged firm while an increase i n si results only i n costs because it does not affect the conflict measure 9 . Figure 1.4 is a three dimensional illustration of the relative advantage of debt d  changes w i t h changes i n si and c. T h e figure illustrates how the advantage from leverage is reduced w i t h si, tends to increase at low levels of c, and decrease at high levels of c.  Change i n variance W e analyze the effect of changing the variance of the economyincreasing effort.  T h i s is interesting because there is a common belief that an  increase i n cashflow variance increases the payoff of the equity stake, while reduces the payoff of the debt claim. T h e intuition behind this belief is that equity is like a long call position, which increases w i t h variance, while debt is like a short put position, which reduces w i t h variance. G a l a i and Masulis (1976) show that firms w i t h the same market value but different variance w i l l have different capital 17  structure. T h e high variance firm would have a higher equity stake and a lower debt stake, resulting i n a lower debt to equity ratio. W e analyze a change i n the variance by increasing the value of while decreasing the value of si to si — e^y^,  to Sh + e,  where e is very small. T h i s change  w i l l increase the variance, while leaving the mean unchanged at s = (1—b)sh+bsi. W h i l e such a change does not affect the contribution value of an a l l equity firm V , we find that it does increase the debt face value D, the incentive a e  d  be learned from eqs.  (as can  1.21-1.22), and the contribution Vd of a leveraged firm.  G a l a i and Masulis (1976) assumed an exogenous volatility of cashflow, while i n our framework the cashflow variance is determined endogenously. T h e higher Sh and lower si means that economy-increasing effort is riskier and that makes riskshifting more valuable to shareholders (as can be learned from the higher (9 ). d  T h e increase i n economy-increasing effort E  results i n shareholders increasing  s  the incentive  However, since shareholders prefer to increase incentives without  actually increasing the payoff to the manager above her reservation wage, they also increase the face value of debt D. T h e result is that contrary to G a l a i and Masulis (1976) the market debt value p  increases.  d  1.4  O p t i o n Grants to Manager  O p t i o n grants and incentives Instead of issuing debt, risk-shifting towards more economy-increasing projects can be induced by granting options to the manager. In this section, we analyze how options can affect the firm's contribution to shareholders portfolio. A t t = 0, shareholders choose the fixed wage wo, the exercise price K, and the percentage of cashflow (5 above the exercise price that they give to the manager (hence, the option incentive (3). T h e exercise price is chosen so that the option is i n the money i n the high payoff state only [cE + siE c  s  < K < cE + ShE ). c  s  At  t = \, given (3 and K, the manager of firm A, decides on the two types of effort. A t t = 2, the cashflows of the firms i n the economy is generated and the payoffs are distributed to b o t h the manager and the shareholders. Figure 1.5 shows the payoff to the manager and shareholders depending on the two different states. A t t = 2, the manager maximizes her utility and chooses the two efforts  U  M  = max  w + 0(1 - b)(s E 0  h  + cE - K) - \{E  2  s  c  18  + E) 2  C  S  (1.24)  Solving for the optimal effort choice we have, E  c  = pc{l-b)  (1.25)  E  s  = 0s (l-b)  (1.26)  h  Thus, similar to the case of risky debt, the firm is subject to the reduced endeavor cost, captured by the (1 — b) factor in eq. (1.26). We refer to the relative effort measure in this firm as 9 and it equals op  Bap = | ? = jb c  (1-27)  c  Note that # = dd, meaning that the conflict of interest between the manager and shareholders when we issue debt or give options to the manager is the same. A t t = 1, shareholders choose an option incentive (3 that maximizes the contribution of firm A to their terminal portfolio value K p , subject to the managers participation constraint. Note that the expected payoff fi to shareholders at t = 1 equals op  s  H, = (1 - b) (s E + cE - (3{s E + cE - K)) + b(cE + s E ) h  s  c  h  s  c  c  t  (1.28)  s  This is because in the high state with probability (1 — 6), the shareholders receive the residual of what is left after payments to the manager, while in the low state, with probability b, the shareholders receive all the cashflow of the firm. Thus, shareholders decide on the optimal option incentive according to the following program: Vop = max (-cE ) + fi + (w - w ) c  s  Q  s.t.  (1.29)  UM  > w.  The optimal exercise K, option incentive /3, and firm vl's contribution to shareholders portfolio can be derived. Proposition 3 The optimal exercise price K and option incentive (3 are K = ( l(l-b)  + bs )  s  lSh  ( L 3 Q )  3 _ 2 (sUl-b) P  -3{(sl  +  bs \  c>)(l-b)) 19  +  lSh  (  L  3  1  )  which results in the firm's contribution  to shareholders portfolio  A{si(i-b)  9  +  b  being  y  SlSh  (si + c»)  The optimal exercise price K increases with  and si, and is unaffected by c. The  option incentive 3 increases with si, decreases with c, and may either increase or decrease with s^.  The firm contribution  to shareholders  portfolio  Vd increases  with Sh and si, and decreases with c. Contrary to the case of debt-in-place, an increase i n c affects only the option incentive Q but not the exercise price K. W h e n options are offered the only conflict to be considered is w i t h the manager. T h i s means that a reduction i n 3 is sufficient to reduce effort and there is no need to also reduce K. Contrary to the case of risky debt, s; plays an important role i n the decision on the exercise price K and the option incentive 3. In the risky debt case, when shareholders set the incentive aid, they disregard the low state (bankruptcy state). Under the option scenario, shareholders care about the low state payoff when they decide on the option incentive 3, and therefore si affects their decision. A higher si increases the expected payoff to shareholders, who increase incentives by setting a higher 3 and a higher K. Thus, the m a i n difference w i t h options compared to debt-in-place is that shareholders consider b o t h high and low states payoff when they make their incentive decision. Similar to the case of risky debt  has a positive effect on the exercise price  K, but contrary to the case of risky debt, an increase i n Sh has ambiguous affects on the option incentive 3. In general 3 can be negatively affected by s  h  (e.g.,  when c = 0 ) . T h e i n t u i t i o n is as follows. A n increase i n s results i n an increase h  i n economy-increasing effort E . Note that as Sh increases, the cashflow i n the s  high state increases by more t h a n that of the low state (both Sh and E  s  increase,  while si is fixed). T h e result is that shareholders may actually want a larger share of the "big pie" i n the high state and reduce the option incentive 3. T h e rest of the comparative static results are straightforward. T h e firm cont r i b u t i o n to shareholders portfolio V increases w i t h s and s/, and decreases w i t h d  h  c. Debt compared to option grants In general, b o t h debt-in-place w i t h equity compensation and option grant incentives seem to have the similar advantage of i n ducing the manager to exert comparatively more economy-increasing effort and less cannibalistic effort. B o t h alternatives result i n the manager ignoring the low state payoff, which reduces the conflict measure. However, it should be clear by now that the two alternatives are quite different. 20  Proposition 4  Option grants to manager can do no worse than  with equity compensation,  in contributing  debt-in-place  value to shareholders portfolio,  i.e.,  Vop > V . d  While the advantage in reducing the conflict measure and the cost of incurring reduced endeavor cost are similar in both the debt and options cases, there is one big difference between the two cases. Debt suffers from the added conflict with debtholders, which results in a commitment problem about the incentive scheme. The fact that the incentive a is decided after debt is in place means that d  shareholders disregard the low (bankruptcy) state payoff when they set incentive a.  18  d  Debtholders are rational and consider this wealth transfer ex-ante. The  result is that shareholders pay for considering only the high state payoff. Contrary to debt, in the option grant case (3 and K are set simultaneously, accounting for both low and high states payoff. Thus, options do not expose the firm to another conflict of interest between debtholders and shareholders. Option grants compared to equity Since options grants are a better alternative than equity compensation in a leveraged firm, we analyze whether they also do a better job than equity compensation in an all equity firm. Proposition 5  Option grants contribute a higher value to shareholders  than equity compensation,  portfolio  i.e., V^ > V . e  We know that options compared to equity have the advantage of inducing economy-increasing effort but also have the disadvantage of causing some reduced endeavor cost. As was discussed previously, the manager ignores cashflow generated in the low state. However, since this results in comparatively less cannibalistic effort and comparatively more economy-increasing effort, shareholders react by compensating the manager with relatively more options than shares. It turns out that the reduced conflict 6  op  (compared to 9 ) together with the higher e  compensation ((3 > a), more than off sets the reduction in economy-increasing effort due to the reduced endeavor cost. The intuition behind this result is rather straightforward. A firm that generates a high variance cashflow contributes more value to shareholders. This is because the high variance is reflective of a high degree of economy-increasing projects. In general, option grants do a better job than equity in inducing the manager to increase variance. Note that the deadweight cost due to cannibalistic projects in the option grant case is eliminated in Note that due to the timing of the model the commitment problem falls under the incomplete contract approach. However, the result would be the same if we had made the standard moral hazard assumption that shareholders' choice of managerial incentive is unobservable. 18  21  the low state. T h e manager does not earn anything i n the low state and a l l of the cashflow belongs to the shareholders. Thus, options reduce the conflict and at the same time reduce the deadweight cost to shareholders due to cannibalistic effort. Figure 1.6 illustrates the relative advantage of options compared to equity for different levels of c and s/. T h e higher is c the more important is to compensate managers w i t h options. T h i s is because the cannibalistic ability of the firm is high and it is important to reduce the conflict between the shareholders and the manager. T h e higher is si, the larger is the reduced endeavor cost, which results i n a lower benefit to shareholders due to option grants incentives. Finally, Figure 1.7 illustrates the cost of the commitment problem that is involved w i t h debt when si = 0. T h e commitment cost is the difference between the relative advantage of options and the relative advantage of debt. W e can see that as cannibalistic payoff becomes high, debt cannot be used i n order to reduce the conflict. In fact, debt does worse t h a n equity compensation i n an a l l equity firm.  1.5  E m p i r i c a l Implications O u r model takes the normative view that due to shareholders' diversification  the o p t i m a l compensation contract should benefit the market as a whole, rather t h a n maximize the value of a specific firm. However, we believe that at present practice this approach is not advanced by corporations.  In broad terms, we  believe that boards of directors pursue policies that maximize firm value, and the effect on other firms is neglected. Thus, we feel that the model is better suited to provide guidelines for future consideration t h a n is able to give testable empirical predictions. In this Section we start by presenting our alternative view to issues i n patent policy and compensation, then we present some evidences consistent w i t h our theory. Patent policy T h e ability to cannibalize on a competitors invention is the reason for the institutionalization of the Patent system. T h e idea being that if one wants to spur innovative activity, it is possible only w i t h the protection on intellectual property, so inventors could appropriate the return on their work. However, protection of intellectual property rights raises complex questions of what constitutes a patent, and what is the protection that should be granted (i.e., patent breadth and patent life). In broad terms, it is not clear how patent policy affects R & D expenditures and the spur of innovation. M a n y people would argue that most of 22  the value of patents comes not from what y o u actually collect from licensing but from the market advantage they secure for the patent owner or licensee. Thus, arguably the real value lies i n a l l the things your competitors could not do, i.e., they could not move into market X, they could not offer feature Y. Under such circumstances, patent policy can actually do the opposite of what it initially was intended for, i.e., increase rather t h a n reduce the ability of a firm to cannibalize. Moreover, w i t h a few exceptions, there is overwhelming evidence that patent protection is unable to prevent i m i t a t i o n and therefore its significance i n providing incentive for inventors is exaggerated. Other factors such as trade secrets or first mover advantage, are often cited as much more important i n providing incentives to innovate (e.gl, Mansfield , 1961, 1985, 1986, Schankerman, 1998). For our purposes, what is more important is the misconception that owners of inventions maximize firm value. If one considers the case of public firms, spurring innovation i n widely held firms can be done by easier and cheaper means t h a n patent policy. One needs only to motivate the decision makers i n the firm level to increase their risk taking initiative. A s shown i n the previous Section, this can be done by options grants. Compensation W h i l e there still exists controversy about the link between executive compensation and stock market value (e.g., Jensen and Murphy, 1990, Haubrich, 1994, H a l l and L i e b m a n , 1998), it seems that the literature is rather clear that the objective of shareholders is to maximize firm value. O u r theory states that shareholders may care more about how a firm's value increases t h a n by how much a firm's value increases. If the firm engages i n cannibalistic projects that increase firm value but reduce shareholder's portfolio value, compensation should actually be negatively related to firm v a l u e .  19  Thus, our model shows that  a high dependency between firm value and manager's wealth is not necessarily optimal. W e believe that managerial compensation policy should not account only for the traditional agency problem, but also account for the agency problem presented i n the chapter. Note also that there is a branch i n the literature that follows the idea that if a particular compensation policy is good for shareholders, we should observe that the structure of the compensation scheme and firm performance are related (e.g., M o r c k et al., 1988, M c C o n n e l l and Servaes, 1990). A g a i n , this line of literature may be at fault w i t h what represents a good compensation policy because the performance of any specific firm is not important Other studies present a rationale to the Jensen and Murphy (1990) result. These include, Aggarwal and Samwick (1999), Narayanan (1985), Stein (1989), Bizjak et al. (1993). Related to our explanation is Aggarwal and Samwick (1999), who consider product market competition among firms. However, none of these papers give justification for a negative pay to performance sensitivity. 1 9  23  for shareholders. Evidence W e show that option grants to C E O s may be a good alternative for creating value for shareholder's portfolio. O u r result is derived from the idea that a convex payoff increases risk taking behavior of C E O s .  Other models  (e.g., Haugen and Senbet, 1981, Carpenter, 2000) show that options may not increase risk taking behavior. W e wish to note that even though these studies have appealing circumstances i n which options do not increase risk taking, the m a i n stream intuition and the empirical evidence is that options increase risk taking behavior. DeFusco et al. (1990), R y a n and Wiggins (2002), and Coles et al. (2002) a l l find evidence that executive stock option plans induce managers to take on more risk. A n empirical fact that is consistent w i t h our theory is the simultaneous tendency i n the last two decades of shareholders being more diversified (Saxton, 2000) together w i t h management being compensated more by stock option grants (Hall and L i e b m a n , 1998, and M u r p h y ,1999). W e suggest that these findings may be reflective of the increased involvement of institutional investors i n setting compensation policy.  Hansen and L o t (1996) provide empirical evidence that  funds tend to combine the ownership of related firms. W i t h i n a specific industry, the ability of the manager to engage i n cannibalistic activity is high. Under such circumstances, option grants do a good job i n inducing manager to concentrate i n economy-increasing projects rather t h a n cannibalistic projects. O u r model predicts that the larger the s and the lower the c, the higher the tendency of shareholders to compensate manager according to firm value (either w i t h equity or options). T h i s is because the compensation scheme is more effective when cannibalistic payoff is comparatively small to economy-increasing payoff.  In our framework, c and s are determined exogenously and represent  the investment opportunity set (i.e., a firm's specialized physical and human capital). G r o w t h companies tend to have more economy-increasing projects, and therefore, represent a natural interpretation to high s/c firms. I n broad terms if s = 0 and c > 0, shareholders would not compensate the manager according to firm value at a l l , because a l l the firm does is increase its value at the expense of other firms. Therefore, C E O s of growth firms should be compensated more according to firm value. T h i s is consistent w i t h the findings of S m i t h and Watts (1992), C l i c h (1991) and Gaver and Gaver (1993) that C E O s compensation i n growth companies is more closely tied to their companies' performance. O n the other hand, Bizjak, Brickley, and Coles (1993) and Gaver and Gaver (1995) found no significant relation between growth opportunities and compensation i n their 24  samples. W h i l e s and c represent the investment opportunity set, we believe that the two types of efforts, E and E , could be denned as R & D and capital expenditure s  c  respectively. W e believe that innovative costs are mostly R & D , while costs that are invested i n mature markets w i l l fall mostly under capital expenditure. F r o m these two observables one could calculate the relative effort measure of the firms in the economy. O u r model predicts that for firms w i t h similar attributes (i.e., same industry, size, etc.), the relative effort should increase w i t h option grants and leverage and decrease w i t h equity compensation. Consistent w i t h this prediction, R y a n and Wiggins (2002) find that options exert a positive influence on R & D while restricted stock exerts a negative influence. E v e n more interesting is Coles et al. (2002) findings that higher convexity i n compensation is associated w i t h relatively more investment i n R & D (very significant) and advertising (significant), while equity compensation (measured by delta) reduces tendency to engage i n R & D , but does not affect advertising. T h o u g h the authors do not attempt to address this difference i n result between R & D and a d v e r t i s i n g , we 20  believe that our model may provide an interpretation for these results. W e believe that advertising expenditure can be either economy-increasing and risky (promot i o n of a new product i n a new market) or cannibalistic and safe (promotion i n a known mature market w i t h many competitors). Under such circumstances advertising should not be reduced when C E O is given more equity compensation since it w i l l result i n a shift from the risky advertisement to the safe cannibalistic advertisement.  1.6  Conclusion Studies that focus on firm value m a x i m i z a t i o n have as an underlying assump-  t i o n either that shareholders are not diversified, or that product markets are completely competitive. In this chapter, we took an alternative view and assumed that shareholders hold the market portfolio and therefore care about the combined value of a l l firms i n the market. W e showed that i f we consider that an opportunity set enhancing project is riskier t h a n the typical project, risk-shifting is a good t h i n g because it makes the manager engage i n more of the projects that increases shareholder's portfolio value. W h e n analyzing how debt may affect the value of the shareholder's portfolio, we show that Jensen and M e c k l i n g (1976) agency cost is only partially a problem. W h i l e the inability of shareholders to  The authors assume that both R&D and advertising are risky expenditures and study the relation between risk taking behavior and compensation. 20  25  commit to the managerial incentive scheme is a cost, the fact that the manager ignores the payoff i n bankruptcy states of the world is a benefit. Thus, the ability of shareholders to transfer wealth from the debtholders to themselves is a problem, but their is no meaning to the cost of t a k i n g on negative NPV  investments.  T h i s is because shareholders are concerned w i t h the expected cashflow of a l l the firms i n the economy, and not the expected value of any specific firm. In fact, i n our framework, the benefit of debt is i n its ability to shift the manager choice of projects to the riskier value creating ones. Options grants that are given to managers are also able to do such risk-shifting.  However, different from debt,  w i t h options the firm is not exposed to the added conflict between debtholders and shareholders. T h e result is that while there is a cost for using debt to i n crease portfolio value, there is no such cost when.using option grants. W e show that option grants are not only a better alternative t h a n debt, but also a better alternative t h a n equity compensation. 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Journal of F i nancial Economics 32, 263-92. 59.  Stein, J . (1989), "Efficient C a p i t a l Markets, Inefficient F i r m s : A M o d e l of M y o p i c Corporate Behavior." Quarterly Journal of Economics 104, 655-669.  60. Tirole, J . (1988), "The Theory of Industrial Organization." Cambridge, M a : M I T Press.  31  Appendix Proof of Proposition 1 A t t = 1 the contribution of a n all equity firm is,  V  =  e  m a x (—cE ) + (1 — a)fi + (w — w ) c  A  0  a  S.t  U  >  M  =  W  max (—cE ) + (1 — a) (sE + cE ) c  s  c  a  =  m a x (1 — a)sE  — acE  s  c  a  — max (1 — a)as (s — jcr) — a c 2  2  a  T h e first order conditions can be written ^ = (1 - 2a)s - 2ac = 0 da which gives the optimal a. 2  2  s  2  a  2 (c + s ) 2  2  Clearly, this first-order condition defines a unique m a x i m u m , as  < 0.  T h e fixed wage w is chosen so that the managers is b i n d t o her reservation 0  wage, i.e., UM = u>-  wo + W -^(E  2  A  1  +E ) = w 2  s  2  w  = w-a(sE  WQ  = w — -a (c + s )  0  + cE )-^(E  + E)  2  s  2  c  s  Substituting a and WQ back into V V  e  + (1 - a) (sE + cE ) + (w - w )  =  -cE  =  (1 - a)sE  =  a(l - a)s - a c •+  =  as - ]-a (c + s )  c  s  - acE  a  2  + \a (c 2  c  2  2  2  2  2  2  32  c  -a (c  2  2  2  0  -f s ) 2  + s) 2  Substituting for a we get 3s  V,= 8 (c  2  4  + s) 2  W e take the partial derivatives of a w i t h respect to c and s.  d  S  (  SC  \  2  2  <9c V 2 ( c + s ) ; 2  s  9 /  2  2  ds \2(c + s )J  (c + s )  2  2  Q  2  sc  \  2  <  (c + s )  2  2  2  2  >  0  W e take the partial derivatives of V w i t h respect t o c and s, e  3 (  35 .  \  4  3  acV8(c + s ) ; 2  3 /  3s  ds \8(c  S  2  + s )J  + s ) 2  2  2  2  (c + s )  4  2  2  2  3 3 2c + s  \  4  c • •  4  4 (c  2  2  2  > 0  Proof of Proposition 2 A t £ = 0 the shareholders choose the o p t i m a l debt level,  V =  m a x (p +  d  d  Vid) + {w -  w) 0  s.t. UM  > w.  Note that the participation constraint of the manager is actually composed of two parts:  (1)  :  w + a (l - b)(s E + cE - D) - ±(E + E ) > w  (2)  : cx (l-b)(s E  2  0  d  d  h  h  s  + cE -D)-^(E  2  s  2  c  c  +  E )>0 2  T h e first condition is the overall wage of the manger which has t o be more t h a n or equal t o the competitive wage. T h e second condition must hold i n order that the manger engage i n any effort at all. I n other words, the manager has the option t o engage i n no effort and earn only the fixed wage WQ if engaging i n effort produces a negative sum. Thus, the best shareholders can do is to set to = w 0  and eliminate the first condition. Note also, that the t i m i n g of the payment of the fixed wage (either £ = 0 o r £ = l ) i s not important and has no affect o n any of the results. 33  Thus, we are left w i t h the following problem. V  d  =  max (p + V )  =  m a x (1 - b)D + b(cE + s E ) + (-cE ) + (1 - a )(l  =  m a x (1 - b)D(l -l  =  m a x (1 - a )(l  d  ld  c  t  + ) h  c  C  C  c  - b) + a D(l  d  h  + cE - D)  s  c  -fe).)+ (1 - a )(l  d  - cE a (l  s  - b)(s E  d  + E (b - 1 + (1 - a )(l  ad  - b)s E  d  a  d  - b)s E h  - b) + bs E  d  t  s  Substituting for E , E , we have s  c  V = m a x a (l - a )(l - bfs  2  d  =  d  a (l - bfs  Note that  - a (l - bfc + a D(l 2  h  2  h  d  h  =  J  d  b) s Sl  - b) + a b(l - b)sis  2  2  d  - b) + a b(l -  2  d  - cx (l - bf ( c + s ) + a D(l  2  d  d  d  d  h  >0  2  T a k i n g the first derivative we get, dV  (1 - b)s  D bs s a (l - b) + + a ( l - fe) + ^ ' 2f>»+c?) " ' 2 ( 4 + c2)  2  d  -  00  n  h  t  d  2 ( 4 + 2) C  2  (1 - 6 ) 4 +  D  2 (s + c ) 2  h  d  2  2  +  h  +c ) 2  Thus, the first derivative is always increasing w i t h D. T h i s means that the shareholders choose the m a x i m a l value of D subject to the participation constraint of the manager. a (l-b)(s E d  a (l d  h  + cE -D)  s  =  c  - b)(a (l - b)s + a (l - b)c - D) 2  d  2  h  d  D  = =  ^(E + E ) 2  a  2  ^ ~ ? (  c  + s )  +  sl)  b  ^ ( c  Note also that the optimal D must lie i n the range of cE +siE c  s  2  2  2  h  < D <  cE +ShE . c  s  T h i s means that cE + siE  s  <  D <cE  a (l - b)(c + s )  <  D < a (l - b)(c + s )  c  2  d  lSh  c  + sE h  s  2  2  d  T h e right hand side is easily verified. However, the left hand side inequality holds only i f c +sis 2  < | ( c + s ) - T h i s is true only i f si > 2  h  2  1). 34  c +2sis .(Assumption 2  h  h  s  + bs E t  s  Substituting D into the equilibrium OJ^ of t = 1, we have  2(1 - 6) ( s + c ) 2  (4(1 - 6) 2(1 - 6)  (4(1  3  4  d  2(1 - 6)  2  fee )  (4 + c )  2  a,  2  2  (4 + c )  4  +  - ft) - 6c ) 2  (l-6)(4 + c ) 2 / 4(1 -fe)- be 2  2  2  Old  3 V(i-&)(4 + c ) 2  and this gives the optimal debt level  D  - 6) -  =  w  Since the debt face value must be positive we must have 4(1 — b) — be > 0, 2  o  ^ > V^i-fc) (Assumption 2)- T h e higher s is compared t o c, the more are  r  h  shareholders willing t o take o n the bankruptcy probability and its related costs. In order t o get the contribution of firm A t o shareholders portfolio we substitute the o p t i m a l D and a  d  V  d  =  a (l - b) s - CY (1 - b) ( c + 4) + a D(l  =  a (l - b) s\ - o? {l - b) ( c + 4) + a (l - b )  2  2  2  2  2  d  d  2  2  d  2  d  -b) + a b(l d  4 { 1  d  ~  6  )  "^  C*d(l 3  « d ( l - fe)  (4(1 - 6)4 - 2 (4(1 - b) - be ) - be + 3bs ) 2  2  lSh  3  QJd(l - fe) (2(1 - 6)4 + fee + 3bXs s ) 2  t  h  3  2(4(1  —  -  9(4 + ^) 2  W e take the p a r t i a l derivatives of D w i t h respect t o Sh, si, a n d c. 3 5  b) s Sl  h  + a b(l d  b)s  lSfl  9  ds \  W - » > - * Y _ 3  h  a  /4(i-6j-6c  5s,  V  |  _  6 ) > 0  2  0  3 - 6) - bc \  d (4(1 dc\  S k ( 1  3  J  26c  2  y  3  3  W e take the partial derivatives of ad w i t h respect to  d 5s  (2s\{l-b)-bc \  _  2  4^  V3(l-6)(4 + c ); / 2 4(1 ~ ^ ~ ^  a  2  3 " ( 1 - 6 ) (4 + c )  2  fc  c  si and c.  2  \  =  2  n  V3(i-6)(4 + c )7 a (2 4 ( 1 - 6) - 6c \  4  dc V 3 ( l - 6 ) ( 4 + c ) 7  3 (l-6)(4 + c )  a,  2  a  2  2  2  W e take the partial derivatives of Vd w i t h respect to 2  fc  (4 + c ) 2  Sft  (44(6  2  (4 +  c )  t  2  a \9 9  s , and c.  / 2 (4(1 - 6) - 6c ) (24(1 - 6) + 6c + 36s,s )'  5  2  < 0 2  2  - 26 + 1) + 8 s £ c ( 6 - 26 + 1) + 3 6 s , s £ ( l - 6)+ 2  2s bc (2b - 1) + 36c A  (34 - c 6 - 2s^6))  2  2  h  T h e numerator is made of 5 terms.  2  T h e first, second, and t h i r d t e r m are  positive because 6 — 26 + 1 > 0 and 1 — 6 > 0. T h e fifth t e r m is positive because 2  by A s s u m p t i o n 2 we mush have ( ^ )  2  >  If we divide the quantity i n the  parenthesis by c we have, 2  s  <?  2  h  2  h  2  c£-»-^»3r^-»-2rh M u l t i p l y i n g the right hand side by (1 — 6) we have 36 - 6(1 - 6) - 26 = 26 - 26 = 26(1 - 6) > 0 2  2  Thus, the fifth term is always positive. O n l y the forth term 2s fec (26 — 1) can be negative when 6 < 0.5. Note, 4  ft  however, that adding the second and forth t e r m is always positive, since  36  8 s c ( 6 - 26 + 1) + 2s bc (2b 3  2  2  4  h  A g a i n , since ( ^ ) >  - 1) = 2 s c ( 4 6 - 8b + 4) + 2s bc {2b - 1). 3  2  2  A  h  ^  2  2 s | c ( 4 6 - 86 + 4) + 2s bc (2b - 1) > 2s c —^-(46 - 86 + 4) + 2s bc (2b - 1) 1—6 2  2  4  4  h  2  A  /l  h  T h e expression is positive i f Y Z £ ( 4 6 - 86 + 4) + (26 - 1) > 0, or i f 2  (46 - 8 6 + 4) + ( 2 6 - l ) ( l - 6 ) 2  26 - 56 + 3 2  >  0  >  0  T h i s expression is negative only for 1 < 6 < 1.5, which means that for 0 < 6 < 0.5 the expression is always positive.  d  (2 (4(1 - 6) - 6c ) (24(1 - 6) + 6c + 2  3bs )  2  ds \9  lSh  (4 + c ) 2  t  = l^ - ^- ^ w^ > 1  b  b  b  0  )  d (2  (4(1  - 6) - 6c ) 2  (24(1 -  (4  6) + 6c + 3 6 s ^ ) 2  dc \9 +c ) 2 2 4 4 ( - 6 + 36 - 2) - 246 c - 6 c - c — ^—, 9 + c ) 2  2  =  2  (4  2  36s 4  4  z  2  n  <0  T h i s expression is always negative because for 0 < 6 < 1, (—6 + 36 — 2) < 0. 2  Change i n variance Substituting  + e a n d si —  for Sh a n d si respectively  i n eq. (1.23) a n d eliminating the e terms, we have, 2  2 ((4 + 2s e)(l h  d  "  - 6) - 6c ) (24(1 - 6) + 6c + 3bs 2  2  lSfl  + 3bs e + s e(l - 6)) t  h  (4 + c )  9  2  F r o m comparing this w i t h eq. (1.23), we can see that shareholders are better off w i t h the increase variance. P r o o f of P r o p o s i t i o n 3 V  op  s.t. UM  = m a x (-cE ) c  . > w. 37  +p  s  + (w-  w) 0  Note that the similar t o the case of debt, the participation constraint of the manager is actually composed of two parts: (1)  :  w + P(l ~ b)(s E  (2)  : (3(l- )(s E  0  h  b  h  + cE - K ) - ±(E + E ) > w 2  s  S  +  s  2  c  cE -K)-±(E +E )>0 2  c  Thus, the best shareholders can do is to set w  0  2  c  s  = w a n d eliminate the first  condition. Thus, we are left w i t h the following problem.  Vop = m a x (~cE ) + fi c  3  s.t. > w.  UM  W e first concentrate on the m a x i m i z a t i o n of Vop, Vop =  max(-cE )  Vop =  m a x -fl(l-b)cE  c  + (l-b)((l-fl)(s E h  + cE )+flK)  s  + (l-b){l-fl)s E  c  + b(cE + s E )  c  h  s  Substituting for the optimal choices E ,E , C  +bE  c  Sl  s  +  l  (l-b)flK  we have  S  Vop = m a x - {3\l - b) c + (1 - b) ((3 - {3 )s + (1 - b)f3K + b{\ 2  2  s  2  2  2  b)0 s Sl  T h e first order conditions can be written  ^  = -2/3(1 - b) c + (1 - 20){\ - b) s + (1 - b)K + 6(1 - b)s 2  2  2  2  lSh  =0  which gives the optimal f3. Clearly, the first-order condition defines a unique m a x i m u m , as  < 0. Note also that the contribution value to shareholders  always increases w i t h K. W e have (s (l-b) 2  0=  Note that the option quantity  + bs ) + K lSh  2 (4 + c ) ( 1 - 6 ) 2  depends o n the exercise price K. Thus, the  o p t i m a l combination of (fl, K) is that which involves that highest level of K possible, satisfies the above condition, and binds the manager's payoff to her reservation wage. 38  h  W e solve for the m a x i m a l value of K, which binds the manager t o the reservation wage.  0(l-b)(s E h  + cE -K)  s  =  c  0(l-b)(0(l-b)s +0(l-b)c -K) 2  \(E + E ) 2  =  2  2  ~ \c  p2{l  b)  + s)  2  2  Note that similar t o debt, we must verify that the optimal K is i n the range of cE + siE < K < cE + ShE . T h i s means that c  s  c  s  cE + s E  s  <  K < cE + s E  0(1 - b)(c + s )  <  K<P(l-b)(c  c  t  2  lSh  c  h  s  + s)  2  2  T h e right hand side is easily verified since. However that the left hand side inequality holds only i f si > c + 2siSh (Assumption 1). 2  Substituting for K i n the equilibrium 0 we have, (s (l-b)  + bs ) + ^ ( c  2  lSh  2  + s) 2  2 (s + c ) (1 - b) 2  (sl(l-b)  0  3 4  2  (si +  2  + bs ) +0 lSh  C  2)  (1 - b) 4  (sKi-by+bstah) 2 (s +rf)(1 - b)  P  2  h  P  =  2 fsl(l-b)  +  3\(sl  c*)(l-b))  +  bs \ lSh  and this gives the exercise price,  K  =  4 ( 1 - b) + bsis  h  These unique 0 a n d K lead t o a n equilibrium contribution t o shareholder's portfolio of 3 9  V  op  - 0 (1 -  =  2  2  2  2  2  0 (1 - b) c + (1 - b) (/3 2  ^  l  2  ~ 3  b  2  2  2  i_- (4(1 -b) 0 )4 + ^—92  ~3  (-3/3(1 - b)c + 3(1 - 6)(0 - 0 )4  )  b)0K + 6(1 - b)Pa a  b) c + (1 - b) ((3 - (3 )s + (1 -  2  2  t  + ba.sh) + bp s (l Sl  + (4(1 - 6) + bs ) lSh  0 ( 1 - b) (-3/3(1 - 6) ( c + 4) + 3(1 - 6)4 + (4(1 - 6) + bs ) 2  lSh  3  0 ( 1 - b) ( - 2 (4(1 - 6) + bs )  + 3(1 - 6)4 + (4(1 - 6) + bs )  lSh  3  h  lSh  + 3b(3s ) lSh  +  3bp s )  \-"n\~  "J  4 (4(i - 6) +  (4 + c  9  bs y lSh  2  )  W e take the partial derivatives of K w i t h respect to Sh, si, a n d c.  ^(4(i-6  d  ds \  3  h  d  6^n  ) +  f(s (l-b)  ; lSh  5si V  3 i 5 / l  )  +  b  s  i  >  0  1 ~bs > 0 3  =  7  5 / A ( 4 ( l - 6 ) + 6s 5c V 3(1-6)  b  3  + bs )\  2  i  =  h  )\ J  °  W e take the partial derivatives of 0 w i t h respect to Sh, si, and c.  a  / 2 4 ( l - 6 ) + 6s;g \  ds  h  \3  (4 + c ) (1 -  =  6) )  2  2 -2s c  "  (-1 + 6) (s + c ) 2  c ( l - 6 ) + 6s, 2  g / t  2  fe  3  3 2  2  h  22  U s i n g the fact that s\>c  + 26s c + 6^4 - 6s,c  2  fe  (1-6)  2  2  (c -4)  (4 + c  2  2  )  2  + 2sis , we know that we must have h  dp 0S  fc  4 5 (c (l-6)-64) 3 (1 _fe)( 2 2 ) 2 2  f a  <  S +  C  In general this can be negative when 6 is relatively high, a n d c is relatively low t o  Si.  40  h  + 3b(3 s )  0 ( 1 - b) 2  Sl  Sl  h  h  d f2s (l-b)  + bs \  2  h  2  lSh  h  3 ( l - 6 ) ( 4 +c )  dsi \3(sl + c*)(l-b)J d (2sl(l-b)  s  h  2  + bs \  4  lSh  ^ ,  W e take the partial derivatives of  o  c  w i t h respect t o Sh, si, a n d c.  bs y  U(si(i-b) + ds \9 (si + c )  lSh  2  h  8 -a  fc  ^U  0c ^ 9  2  — >  l  0  „  2  ) \  8 ,  2  | f l h  (4 + c )(i-fe)  J  (4 + c ) ( l - f e )  J  2  A  ;  h  / 4 ( 4 ( 1 - 6 ) + 6a A  fes,c  2  S / l  ;  9  5  , „ ,v , , 4 ( 1 - f e ) + 2 c ( l - f e ) + ( a ( l -fe)+ 6a,) * \ (4 + c )  2  ^  =  =  (  „ ^  (  1  - 9 ' * ^  -  1  6  )  +  -  6  )  ^ (4T^) )  +  6  " ) ^  ?  >  0  < 0  Proof of Proposition 4 Let A = (1 - 6)4, £ = 6c , and C = 6sjs . 2  h  Substituting A, B, a n d C into Vd and V^,, we have 2 (A - B) (2A + B + 3(17) Vd  V  m  =  9  =  (4 + c  2  )  4 (A + Cf 9(4 + c ) 2  W e calculate the difference V — Vd op  22(A °  P  d  " 9  + C)  2(A-B)(2A  2  (4 + c )  9  2  + B + 3C)  (4 + c  2  )  T h i s difference is positive if and only if,  2(A + C)  2  2 A + 4 A C + 2C 2  2  AB + AC + 2C + B + 3BC 2  2  > (A - B) (2A + B + 3 C ) > 2A + 2  > 0 41  AB +  3AC-2BA-B -3BC 2  Note that A — (1 - 6)4 > 0 and B = be > 0. T h e only variable that m a y be 2  negative is C =  because  bsiSh  can be non-positive. However, their is a limit  si  on this negativity of s (Assumption 3). Thus, the lowest possible value for s/ is t  —j^i w h i c h gives a lower limit for C of C = —be = —B. Thus substituting for 2  this m i n i m u m value of C we have, AB + AC + 2C + B + 3BC 2  AB-AB  =  2  + 2B + B -3B 2  2  = 0  2  Thus, for any other value of si higher than the lower limit, we have a positive difference Vgp — ViP r o o f of Proposition 5  8(c + s ) 2  2  A(s (l-b)  +  2  h  op  9  lSh  (s + c )  4(s  h  2  ( (l-b)  + b ))  2  Sh  9  Sl  (s + c ) 2  4  2  js s)  2  h  H 4  T h i s means that V  bs f  2  + c) 2  > V i f and only i f  op  e  >  8(c + s ) 2  2  3s  2  >  8(c + s ) 2  2  or 32s (c + s ) 2  2  2  l  54«  2  +  c (324 2  27s ) 2  >  27s (4 + c )  >  0  2  Since Sh> s, this condition is always satisfied. 42  2  Shareholders set Shareholders  manager's  issue debt at  fixed wage w  face value D  and incentive a  t=0  0  Managers choose  Payoff is  the effort  distributed  level E  d  and E  c  t=1  s  =2  t=3  t  Figure 1.1: T i m i n g of the debt model: A t t = 0 diversified shareholders issue a face value D of risky debt. T h i s debt is valued i n the financial markets at a price of p  d  and the proceeds  of the debt issue are distributed to shareholders as dividends. A t t = 1, investors choose the fixed wage WQ and the incentive share a that they give to the manager. A t t = 2, the manager of firm A decides on the two types of effort E  c  and E . s  A t t = 3, the cashflows  of the firms i n the economy is generated and the payoffs are distributed to the manager, shareholders, and debtholders.  43  Shareholders payoff  Debtholders payoff  s + cE  Figure 1.2:  c  Debt model: the t = 3 cashflow distribution t o manager, shareholders,  and  debtholders i n the 2 possible states, bankruptcy (probability b) and solvency (with probability  1-6).  44  1.8 J,  1.7  Sj  =0 =0.15 I Sf, = 0.30 ls  h  1.6 1.5  K 1.3  -.  - -- '  ^  1.2 1.1  0.9  3  0.1  0.2  0.3  0.4  cls  0.5  0.6  h  Figure 1.3: T h e relative advantage of having debt-in-place compared to an a l l equity firm for different levels of cannibalistic effort payoff c and economy increasing effort payoff i n the low (bankruptcy) state si. T h e numeraire is the high state economy increasing effort payoff Sh  =  1.  45  Figure 1.4: A 3-dimensional illustration of the relative advantage of having debt-in-place compared to an all equity firm for different levels of cannibalistic effort payoff c and economy increasing effort payoff in the low (bankruptcy) state Sj. The numeraire is the high state economy increasing effort payoff Sj, = 1.  46  S h a r e h o l d e r s payoff sE h  + cE  s  c  - P(s E h  + cE  s  c  - K) =  M a n a g e r payoff  fi(s E h  s  •+ cE  c  -  K)  0 Figure 1.5: O p t i o n grants model: the t = 3 cashflow distribution to manager and shareholders, i n the two possible states, the down state (probability b) and up state (with probability  1-6).  47  Figure 1.6: T h e relative advantage of having option grants compared to an a l l equity firm for different levels of cannibalistic effort payoff c and economy increasing effort payoff i n the low (bankruptcy) state s,. T h e numeraire is the high state economy increasing effort payoff Sh  = I-  48  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  cls  h  Figure 1.7: T h e commitment cost: the difference between the advantage from options grants and the advantage from debt-in-place represents the commitment cost. T h e commitment cost increases w i t h the magnitude of cannibalistic effort payoff c/sh-  49  C H A P T E R II A RATIONALE FORTHEMARKET M A K E R  2.1  STRUCTURE  Introduction A r o u n d the world exchanges have been converting their trading systems from  the t r a d i t i o n a l call market to continuous trading alternatives. B y allowing for continuous trade exchanges are presumably improving trade by increasing the liquidity of the securities traded. G i v e n that one aspect of liquidity is the time involved before an asset can be sold or bought, this progression towards continuous trading seems appropriate. W i t h i n the class of continuous markets, trading can be accomplished using different mechanism for continuous trade. In broad terms, continuous trading can be accomplished either by computer based limit order books, or by market makers (designated dealers) that serve as intermediaries. T h i s raises a few questions: First, why is it that a l l recent changes i n exchange trading mechanisms have been towards the computer based limit order book trading alternative and not the market maker trading alternative (e.g., France, Spain, Italy, Israel)?  Second, i f the reason for choosing the computer  based alternative is simply the cheap processing of trade that computers are capable of today, then why do market makers still exist i n developed markets such as the N Y S E and Nasdaq? In other words, what makes the market makers presence valuable when continuous trading can be accomplished by other cheaper means? Market microstructure research has provided i n the last two decades major insights to the behavior of securities markets.  However, a criticism of many  of the market microstructure models is that they give little or no guidance on welfare effects. M a u r e e n O ' H a r a (1999) claims that even Christie and Schultz's (1994) well known findings on the collusion of Nasdaq market makers may merely be a transfer problem: the dealers took from the traders, and the lawyers took from the dealers, but on balance the pie was still the same.  Thus, although  microstructure of markets has real effects on market participants, the overall welfare effect remains unclear. T h i s lack of knowledge is largely due to the set up of many microstructure models, i n particular the behavior of liquidity traders who are presumed to trade no matter what. T h e objective of the current study is to investigate the welfare effects of choosing a monopolist market maker trading structure over a competitive structure such as a limit order market.  Competitive limit order markets feature trade  between public investors without dealer intermediation, while i n the monopolist market maker structure, a market maker serves as an intermediary through  50  which a l l trades i n the market are facilitated. T h e welfare comparison is done i n a market w i t h symmetric information to a l l investors. W e assume that the market maker serves only as an intermediary who buys shares from the seller and sells t h e m immediately to the buyer.  Thus, we model the economy i n a  setup which is only loosely connected to most of the microstructure literature. M o s t market microstructure literature focuses on informational issues . 1  These  models (e.g., K y l e (1985), Glosten and M i l g r o m (1985)) assume different inform a t i o n structures and preferences for the different agents i n the economy, i.e., the informed investors, the market makers, and the liquidity traders. However, M a d h a v a n (2000) stresses that the ultimate decision on market structure is likely to be decided on the basis of factors that have less to do w i t h information t h a n many economists apparently believe. Following this thought, we abstract from informational differences between investors and we show that even under the homogenous information and preferences assumption, the market maker may still have a positive role i n financial markets. In this study we propose and develop a theory i n which the relative advantage of having market makers depends to a large extent on the importance of providing more liquidity i n the market. T h e m a i n insight that is provided i n our model is that a market maker can reduce what we call liquidity risk by solving a free-rider problem that occurs i n incomplete markets that have a competitive limit order book market structure. W i t h o u t a market maker, the investors i n the economy do not take into account their own effect on liquidity when deciding on the amount of trade.  D o i n g so is rational since each investor is "small" and assumes that  her own trade has no effect on t o t a l market liquidity. E v e n though i n our model greater volume increases liquidity, each individual investor prefers to free-ride on the volume generated by the other investors and the result is that less volume is generated. T h e individual investors do not generate more trade even though if a l l the investors i n the market had done so, they would have made the asset more liquid, which might have increased a l l of the investors' utility. T h e investors simply conjecture the aggregate volume and use their conjecture as an exogenous variable i n their maximization problem. A market maker that provides a pricing schedule to the market and exerts search and promotion costs to increase trading volume can solve the free-rider problem. T h e market maker facilitates a l l trade i n the economy and guarantees a reduction i n liquidity risk. In return the market maker collects a fee. T h i s formulation serves as a way to endogenize the volume consideration i n each of the individual investors' maximization problem. Thus, similar to other public goods, an institutional arrangement might achieve a better Madhavan (2000) and Biais et al. (2000) provide surveys of the microstructure literature.  51  equilibrium outcome. A l l o w i n g for competition among market makers shows that the market makers have their own free-rider problem. E a c h of the market makers prefers that the other market makers pay the search and promotion related costs that reduce future liquidity risk. In other words, a l l the market makers benefit from the fact that there is more volume i n the economy, while their cost is only dependent on their own excess volume generated. T h e result is that we have a free-rider problem from the market makers perspective. A s more market makers compete i n the market, each market maker can commit to a relatively lower t o t a l excess volume which results i n a double free-rider problem. Thus, while a monopolist market maker can endogenize the volume decision completely, a competitive structure cannot. I n fact, a high degree of competition completely eliminates the ability to reduce liquidity risk and improve the welfare of investors. C o m p a r i n g the monopolist market maker and the competitive limit order book it is found that the initial endowments of the investors play an important role. Depending on the opportunity for improving risk sharing i n the economy, a monopolist market maker structure can be efficient. B y dividing the investors into two groups representing the ordinary shareholders who hold diversified portfolios and controlling shareholders who have a major stake i n a company's equity (undiversified investors), we are able to derive an empirical hypothesis concerning the o p t i m a l market structure choice. W e find that market makers can increase welfare i n economies where there is less opportunity for improving risk sharing. In these economies, where most of the equity is held i n diversified portfolios, the ability to improve risk sharing is small, and liquidity may be scarce. T h i s makes the role of the market maker i n providing a pricing schedule important because it endognizes the liquidity risk for investors. O n the other hand, i n economies where most of the equity is held by investors who hold undiversified portfolios, the ability to improve risk sharing is high. In these economies, the role of the market maker becomes unnecessary, and the payment of fees for their services is welfare reducing for the investors. T h e major contribution of this chapter is to provide a welfare analyses of two different market structures. T h i s analyses is done by considering two important features of financial markets: their ability to provide liquidity and their ability to increase risk sharing between investors. O u r approach is related to several strands of the economics and finance literature.  In the field of asset pricing, a growing b o d y of research serves to  demonstrate the importance of liquidity as a factor i n determining expected returns for trading on securities. Theoretical works i n this area include C o n -  52  stantinides (1986), Heaton and Lucas (1996), Vayanos (1998) and Vayanos and V i l a (1998). Similar to what is done i n those models, we model liquidity exogenously. However, we base our analyses differently by building on a relation between volume and transaction costs.  Moreover, we do not concentrate on  pricing issues, but rather analyze investors welfare to obtain market structure implications. There exist a few studies that deal w i t h the liquidity externality. These include A d m a t i and Pfleiderer (1988), Glosten (1989), Foster and V i s w a n a t h a n (1990), Brusco and Jackson (1999). In these models either the investors choose on their own to concentrate their trades on a single market at a single point i n time or they do so w i t h the help of the market makers.  T h e idea is to benefit from  the externalities generated by other traders. However, different from our study, all of the above studies build on heterogeneity i n either preferences or private information. Thus, although their theories present insights to circumstances i n which liquidity becomes an important factor, there results are dependent on the underlying assumptions about private information. Dealing w i t h the liquidity externality without adverse selection are Pagano (1989) and Economides and Siow (1988). However different from what is done i n this study, Pagano (1989) addresses the issue of concentration and fragmentation of trade across markets and not their structure, while Economides and Siow (1988) study the o p t i m a l number of markets for an asset. O u r free-rider problem occurs because of non-cooperative behavior between investors i n large populations. T h i s relates our chapter to a subset of the economic governance literature that show how such behaviors offer profit opportunities for individuals who can solve them (e.g., D i x i t (2001)). In our study, a market maker, who is motivated by private profit rather t h a n by social welfare, is able to improve welfare where investors fail to cooperate. Finally, our results bear some resemblance to the literature on welfare effects i n incomplete markets. For example, similar to E M (1995), we show that by introducing market makers we can achieve an equilibrium which may either make everyone better off or make everyone worse off. T h i s is similar to the results i n incomplete markets w i t h more t h a n one consumption good. T h e rest of the chapter proceeds as follows. In section 3.2 the model is developed and the differences between the limit order book structure and the monopolist market maker structure are formed. In section 2.3 we solve the model numerically and compare the two market structures. W e show how risk sharing and liquidity risk play a role when deciding on the o p t i m a l market structure. In section 2.4 we analyze the relaxation of the monopolist market maker assump53  tion. I n section 2.5 we discuss the empirical interpretation of our model and its implications for public policy. Finally, i n section 3.5 we conclude.  2.2  The Model  W e consider a multi-period economy i n which investors may experience a liquidity shock, which may force them to buy (or sell) the risky asset i n the market. However, i n order to b o t h illustrate and analyze the model, it is sufficient that we model only two dates (t = 0, t — 1) of the economy. Investors have concave u t i l i t y functions and are allowed to trade i n a riskless asset ("cash") and one risky asset ("stock"). There are two types of aggregate investors distinguished by different endowments.  For ease of illustration, the investors have the same  C R R A utility function. T h e investors act as atomistic price takers m a x i m i z i n g the expected utility of terminal wealth. T h e riskless rate is exogenous, and there is unrestricted borrowing and lending at the riskless rate.  For simplicity, we  assume without loss of generality that the riskless rate is zero. There are no transaction costs or taxes, and asset shares are infinitely divisible.  Figure 1  illustrates the value of the stock at t = 1, as perceived by investors at t = 0. A t t = 1 the economy may be i n one of two states, either the good state w i t h probability TT or the b a d state w i t h probability (1 — TX). T h e value of the stock i n each of the two states is stochastic, however, it is known to have i n expectation a value of A or B, i n the good or the bad state respectively. T h e actual distribution of values i n b o t h states depends on the volume generated at t = 0. B y this we capture the idea that there is uncertainty about the actual transaction price at t = 1 and that this uncertainty depends on the liquidity of the market. In other words, selling (or buying) stock at t = 1 involves transaction costs, which are volume dependent.  W e assume that the higher the volume of trade at t = 0,  the more liquid is the stock, and therefore, the less uncertain the investors are about the t — 1 price of the stock.  Figure 1 illustrates how volume affects  future uncertainty by reducing the variance around the A and B values. M o r e specifically, a higher volume at t = 0 reduces the mean preserving spread (hence, M P S ) i n b o t h possible states of the economy. In the extreme case, when volume is infinite (representing a very liquid market), the variance goes to zero, and the t = 1 price is either A or B. T h i s modeling can also be interpreted as denning two sources of risk, fundamental risk and liquidity risk. T h e fundamental risk is embodied i n the difference between the values of A and B, while the liquidity risk is embodied i n the M P S . 54  L i q u i d i t y and the feedback loop T h e simplest approach of modeling microstructure effects is by allowing for unpredictable pricing errors that reflect frictions such as bid-ask spread. T h e volume of t = 0 is a measure that quantifies these unpredictable pricing errors. W h e n a stock is heavily traded at t = 0, investors are able to sell (buy) the stock close to its fundamental value. W e assume that t = 0 trades may be subject to either a positive or negative liquidity shock at t = 1. Under such circumstances, high volume at t = 0 means many buy and sell orders at t = 1. D u e to the law of large numbers, whether you are hit by a positive or negative liquidity shock at t = 1, your search related costs are small and you can transact at a price close to fundamental value. O n the other hand, a low volume stock has less buy and sell orders submitted at t — 1 and higher search related costs, which results i n more liquidity risk. T h i s means that its market depth is small and that there is more uncertainty about the t = 1 price, resulting i n a higher bid-ask s p r e a d . 2  3  A s an illustration of our definition of liquidity  risk, we would expect that at any point of time, the liquidity risk embedded i n a heavily traded stock such as I B M would be small compared to a more lightly traded stock such as M a y t a g . Empirically, one would usually observe low bid-ask spreads for I B M compared to those of M a y t a g . W e should emphasize that the idea of using volume as a variable that affects future pricing is not a new one. In the market microstructure literature, b o t h inventory and information models predict that volume affect prices, but for different reasons. In the inventory models, higher volumes i m p l y a shorter holding period for market makers and hence lower inventory control costs. In the information models, t o t a l volume acts as a signal about future value and causes a revision i n beliefs. L i q u i d i t y has been long recognized as an important factor i n determining asset prices. Keynes referred to an asset as more liquid t h a n another i f it is more certainly realizable at short notice without loss. T h i s definition suggests that the liquidity of an asset relates to the future uncertainty of an asset's future price. Figure 1 builds on the idea that t h i n markets increase future uncertainty because investors are less assured that their future transaction can be absorbed without adverse price changes. In A p p e n d i x A we show how the magnitude of a future liquidity shock i n the economy can reduce current volume. Other studies have trading volume and absorptive capacity of the market feed positively on This is developed theoretically by Duffle et al (2000), who show that bid-ask spreads are lower if investors can find each other more easily. 2  One can think of volume as a measure that quantifies dollar value of trade per unit of time. This means that a high volume stock is subject to a more frequent updating of its bid-ask quotes. Under such circumstances, the distribution of the stock value in next period's good and bad states would be narrower for a high volume stock compared to that of a low volume stock. 3  55  each other. Spiegel and Subrahmanyam (1995), A d m a t i and Pfleiderer (1988), Pagano(1989), and Scharfstein and Stein (1990) analyze the incentives of the traders to cluster and benefit from the additional liquidity they provide to one another. In these models discretion over the time of trade reinforces the concent r a t i o n of volume and liquidity at certain points i n t i m e .  4  T h e relation between  liquidity and the mean preserving spread i n Figure 1 resembles the approach taken by A m i h u d and Mendelson (1986) who considered a problem by arguing that liquidity should be priced. T h e i r argument is that only investors w i t h long horizons would hold illiquid stocks. T h i s w i l l result i n higher required returns and lower prices for illiquid stocks. T h e larger is the M P S around intrinsic value i n the future price of a stock, the lower is the current price for the stock because it is_second degree stochastic dominance inferior to a same expected payoff stock w i t h a smaller M P S . Thus, our modeling approach enables us to price l i q u i d i t y . 5  T h e endowment economy Since b o t h representative investors are homogenous regarding b o t h preferences and beliefs, the only difference between t h e m is their endowments. T h e investors receive different endowments of cash and stock and then trade at t = 0. T h e two aggregate investors' endowment are made up of cash ( C i , C2 ) and stock (Xi, X 2 ) . T h e total supplies of cash and stock i n the economy are C = C\ + C2 and X = X\ + X2, respectively. E a c h aggregate investor is composed of n identical individual investors. T h e endowments of the individual investors are therefore, either ( ^ - ,  for investors of type 1, and (^f, ^)  for  investors of type 2. W e consider the case of n —> 00 as an approximation of the perfect competitive market. T h e budget constraints for the two types of investors are - (d + PXi)  = -(Ci  + PXi)  ,i  = l,2  (II.l)  A t time t = 0 the investors decide on the amounts of cash and stock i n their portfolios. Since we have assumed that the riskless rate is zero, cash does not earn interest. A s was discussed above, the price at t — 1 depends on the volume of trade. For simplification, without loss of generality, the uncertainty can be represented by the extreme values of the distributions around the A and B. W e Coppejans et al (2001) provide support for these models by analyzing data from an automated futures market. They find that (a) volume and liquidity feed on each other, (b) liquidity is a factor in expected returns, (c) increases in liquidity substantially reduce volatility. Note that all of these findings are consistent with Figure 1. 4  The volume-pricing relationship that we model is also justified by an empirical finding. Breen, Hodrik, Korajczyk (1999) found that increasing the magnitude of net turnover during a 5 minute interval by 0.1% of the shares outstanding produces an average incremental price effect of 2.71%. 5  56  denote these extreme values by A, A, B, B_. T h e two spreads are equal and symmetric: _  2rv  A - A - B - a - j ^ where,  V = P \X  t  and  - X \,  (II.2)  a > 0, (3 > 0  t  Pi(A \A) = Pr(A \A) = Pi(B \B) = P r ( 5 \B) = 0.5  The a and /? parameters are positive and represent exogenous parameters of the economy. A lower a and a higher (3 may represent economies where there is more interaction i n the market place between the investors. For example, if many investors i n the economy are connected to the internet and can submit orders to the exchange on line, we would expect that search related costs be lower. Reflecting the effect of liquidity, dollar volume, quantified by price times quantity traded, plays the important role of reducing future spread. N o t e that the a and (3 parameters control the m a x i m u m possible spread, thus even a V = 0 w i l l not result i n an infinite spread. However, as V —> oo, the spread goes to zero. The four possible t = 1 wealth outcomes for the two types of investors are represented by the following formulation: For i -  1,2  State (A)  :  W = ^ (Q + X A) W = -{c  i  where,  V  =  (II.3)  t  + P(X -X ) i  i  + X (A+  a  X  1  i  \P(Xi-Xi)\.  Similarly, the t = 1 wealth for states A, B, B_ can be derived. The limit order book structure Since i n this model investors differ only i n their endowments, interactions between them on the competitive limit order book structure w i l l produce the Walrasian equilibrium. T h e individual investors' maximization problem involves maximizing expected utility assuming that volume is unaffected by the investor's own trade and equals their conjecture that V = VQ. Formally, we have  57  Max =  (WA)  U{Wi)  Max  for i A l t h o u g h the volume V affects the utility and the equilibrium price, each individual investor is very small and takes the volume as an exogenous variable that equals VQ.  In equilibrium the investors' conjecture on volume has to be  confirmed, so we must have P  =  V  =  0  (II.5)  P(V ,Xi) 0  P(V )  xlXi-XA,  0  for i = 1,2  (II.6)  Figure 2 illustrates the relation between volume and the price of the stock. A movement i n the Edgeworth  trade box represents a change i n holdings due to  trade. For example, point J i and I  2  represent two possible i n i t i a l endowments  of the aggregate investors, and the vector going from these two points represent a movement due to trade. Remark  Since both aggregate investors have the same CRRA  utility, they both  wish to trade so that they move to the diagonal of the Edgeworth trade box. T h i s can be understood by the fact that they b o t h invest the same proportion rj of wealth i n the stock n  =  PXi  PX  W  W  2  x  (11.1)  2  PXi d  PX  2  + PXi  C +  (II.8)  PX  2  2  T h i s relation and the clearing conditions lead to X (C X  - Ci + P(X  - Xt)) = (X58  X )(C t  t  +  PXt)  (II.9)  W h i c h can be simplified to  C  Ci  c  C\ +  2  Thus, under homogenous beliefs and same C R R A utility, investors trade such that their portfolio holdings after trade lie on the diagonal of the Edgeworth trade box. Note that the t o t a l volume is simply the amount of cash that is transferred during trade between the two aggregate investors. I n Figure 2, the vector that moves aggregate investors 1 portfolio from point I\ (initial endowments X i , C i ) to the diagonal (with after trade holdings Xi,Ci)  determines the equilibrium  price. Thus, p  Proposition 1  - ^ - v & \ - ^  There is a unique equilibrium  (IL11)  with volume VQ that solves eq.  (II.4) - eq. (II. 6). P r o o f . T h e proof follows from figure 2. Since during trade investors move to the diagonal and trade cash for stock. The equilibrium 9 always lies i n the range where 0 < 9 < 90 degrees. W i t h an increase i n 9 from 0 to 90 we have: (1)  V increases monotonically-and continuously from 0 to a.  (2)  | X i — X\ j depreciates monotonically and continuously from b to 0.  (3)  P = tan(#) increases monotonically and continuously from 0 to J & P = °°-  (4)  B y (1) — (3), each P corresponds to a unique set of V and \X\ — Xi \.  lv  Q  In equilibrium we must have no arbitrage, thus  B < P  <A  or, V  -  B < =  r  \X -X \ 1  1  < A ~  If 9 = 0° t h a n P = 0, meaning that P is lower t h a n the lower bound B_. If 9 = 90° t h a n P = 00, meaning that P is higher t h a n the upper b o u n d A. Thus, by point (3) above, there must exist a unique P that solves for the equilibrium price. B y point (4) above, this unique P corresponds to a unique VQ. • 59  Solving this problem numerically is done by an iterative process. W e start w i t h some i n i t i a l volume V' and plug it i n for V i n the investors' m a x i m i z a t i o n problem of eq. (II.4). F r o m the maximization problem we derive the two F O C for the two aggregate investors. These two F O C together w i t h the clearing condition allow us to solve for Xi,  X, 2  and P. T h e n we derive the equilibrium volume,  V = P \ Xi — X\ \, and plug it back to eq. (II.4). T h i s process is continued t i l l eq. (II.5) and eq. (II.6) are satisfied. T h e equilibrium volume is defined as Vo. A n important issue that all investors are disregarding i n this economy is their own volume effect on future uncertainty. E a c h individual investor's own trade is small. Investors act as though their own trade has no effect on t o t a l volume, therefore, has no effect on the spread at t = 1. Trading more t h a n the amount needed for a movement to the diagonal is inefficient because it w i l l reduce the expected utility of the C R R A individual investor.  T h i s is a classic free-rider  problem. It occurs because each of the two aggregate investors is made of many individual investors. E a c h individual investor prefers to free-ride on the volume generated by the other investors and the result is that there is no more volume t h a n the amount needed for movement to the diagonal. Thus, no excess volume is generated. Volume is like a public good and because each investor's own volume is small, they a l l do not endogenize their effect on total volume. In equilibrium the volume generated is the volume that would have been generated even i f volume h a d no effect on the spread of t = 1. T h e individual investors w i l l not generate more trade even though i f a l l the investors i n the market had done so, it would have reduced the spread at t = 1, and might have increased a l l investors' utility. The investors simply conjecture the future volume and use their conjecture as an exogenous variable i n their maximization problem. In equilibrium, the conjecture must be proven right. The market maker structure W e assume that a l l trades i n the market are done w i t h a risk neutral monopolist market maker, which takes the opposite side of every transaction, b o t h at t = 0 and at t = 1. In order to make those transaction, the market maker has a proprietary search technology that is not available to other agents i n the economy. T h i s puts the market maker i n a unique position to influence the volume of t = 0. T h e market maker can set a pricing schedule at t — 0 that encourages the investors to increase the amount traded. The investors pay a fee of e per share traded, and i n return the market maker is liable for generating a higher volume that reduces future liquidity risk. Formally, the market maker profit function is 7r(V,e) = 2 e | X i - X i | - c(V - V ) 0  60  (11.12)  where Vo is the volume generated i n the economy without a market maker, V — VQ is the excess volume generated i n a marker maker economy, and c(-) is convex cost function. Since the market maker buys the stock from one type of investor and sells to the other type, the t o t a l fee received as income is twice the amount traded multiplied by the fee e. O n the expense side, the market maker bares a cost which is dependent on the excess volume that is generated i n the economy due to the market maker involvement. T h e function c(-) represents the market maker effort which is a function of search and promotion related costs. For example, the market maker can t r y and increase investors trading volume by interacting w i t h them directly, setting up web sites, etc. A schematic description of the derivation of the market maker schedule is illustrated i n Figure 3. In this figure the endowments of the investors and the i n i t i a l volume VQ (volume generated i n a competitive limit order book structure as described above) are kept fixed. T h e two upward sloping curves represent the market maker supply of excess volume V — V , while the two downward sloping 0  curves represent the marginal revenue curves that are derived from the demand of investors for excess volume V — VQ. Note that along the marginal revenue and supply curves the price P is fixed. Definition  The market maker pricing schedule is a relation between price, fee  and excess volume (P, e, V — Vo) that is a direct result of a market maker profit maximization  problem.  T h e market maker pricing schedule is derived to maximize profit subject to the market maker's participation constraint: 2e \X\ — X\ \ — c(V — Vo) > 0. T h e degree of volume chosen for a constant P :  6  d7v(V,e\P)  _  dV  2 £ _ 9 c _ P ~ dV Pdc_  6  ~  ~  (11.13)  2dV  Since c(-) is an increasing convex function, eq. (11.13) is a necessary and sufficient condition for a m a x i m u m and e is an increasing function of ( V — Vo). In figure We assume that the monopolist market maker always prefers to generate excess volume if it is profitable. One could think that a market maker may decide not to exert any effort and just benefit from 2 e ^ | | . Since some volume is going to be generated anyway, it might be more worthwhile than exerting effort and increasing volume. In such a case volume is reduced compared to the limit order book Vo and the regulator may decide to eUminate the market maker position by allowing direct trade. This will leave the market maker with nothing. Thus, if we consider a repetitive game environment, the market maker always chooses a positive excess volume equilibrium if his profit is greater than zero. 6  61  3 we see the interpretation of the pricing schedule. T h e pricing schedule simply connects a l l the marginal revenue and supply curves that hold for the same concurrent P. W e model the cost function for the market maker by c=-(V7  V), 6  0  where 7 > 0,6 > 1  (11.14)  A low 7 means that it is rather costly for the market maker to generate more volume (e.g., high communication costs, low technology), while a high 7 corresponds to a rather developed market where search related costs are rather small. T h e convexity of the cost function is captured by 6. B y specifying the cost function we are able to derive the market maker schedule.  Since b o t h aggregate investors are composed of n investors, the market maker provides the following schedule to each individual investor.  where V is the predicted volume set by the market maker. p  T h e market maker sets a schedule of fee e for different predicted volume V , p  and investors choose a set of (e, V ) on that schedule. T h i s makes V a control p  p  variable for investors. Because investors are homogeneous and because volume of buyers must equal that of sellers, a l l investors choose the same set of (e, V ), p  and i n equilibrium the market maker's prediction on volume equals the actual volume, i.e., V = V = P \Xi — Xi \. T h i s result means that apart from the role p  of searching, and the role of contracting w i t h the different individual investors directly, the market maker's prediction on volume as set i n the pricing schedule becomes a self fulfilling prophecy.  7  T h e fact that the market maker contracts w i t h each of the individual investors, providing t h e m w i t h the schedule of eq. (11.16), enables volume to become an endogenous variable. In other words, a l l of the investors maximize their t = 1 wealth, t a k i n g into account their own effect on volume. T h e market maker schedule results i n equal quantities of trade for a l l the investors due to their homogeneity and the fact that the market maker serves only as a channel for trade and does not hold inventory. Since a l l investors endogenize the volume We therefore simplify our notation and use V (and not V ) throughout the paper. p  62  effect the liquidity risk of t = 1 is reduced. Thus, the market maker part i n the game is simply to interact w i t h a l l the individual investors and provide them w i t h the schedule. B y providing this schedule the market maker is able to endogenize the volume consideration and solve the free-rider problem. It also means that the market maker does not bare any risk by contracting w i t h the investors.  8  T h e two types of investors now have a different budget equation because trade is costly. + PXi)  = ^ [Ci + PXi + e(Xi - Xi)] , for i = 1,2  (11.17)  T h e investors' utility m a x i m i z a t i o n p r o b l e m involves t a k i n g into account the 9  schedule of eq. (11.16), the budget equations, and their own volume effect on the spread. Formally, the investors problem i s , Max  U(Wi)  Max  <  (11.18)  7T,  - (d + PX -e\X -X \)+—[A  -U  i  +  (1-7T)  u U  +  i  Xi\) + —lA  - (d + PXi - e \Xi - Xi\) + ^  1  n  (Ci + PXi-e\Xi-Xi\)  v  1  where, V = P X - X t  2.3  i  (Ci + PXi - e\Xi-  1  2  1 0  {  w  +  a  — [B +  v  —P  -P a  a  , . v  a  ^ [ B n \ 0 + n (^)  P  P  , for i = 1,2  L i m i t Order B o o k Versus Market M a k e r I n this section the two market structures are compared. T h e comparison is  done numerically and details are provided i n A p p e n d i x B . A monopolist market maker structure is efficient i f it is able to increase at least one type of investors The market maker offers to reduce uncertainty for a price. Thus, in some respect the market maker serves as an insurer. However, note that an insurer risk relys on the law of large numbers, while in this game the contract agreement becomes a self fulfilling prophecy. 8  Maximizing the individual investor's expected utility or the aggregate investor's expected utility is the same since the investors represented by an aggregate investor are identical. 9  "Note that &  =  $ ^ . 63  aggregate utility without reducing the other type investors aggregate utility. It can do so i f the advantage of excess volume generated by the market maker increases the utility more t h a n the cost i n utility due to the fee p a i d to the market maker. One can think of the limit order book structure w i t h collusive behavior of a l l individual investors as the first best solution. If a l l individual investors i n the market would have generated more volume and endoginized the volume choice, t h a n a market maker structure could not have been efficient. However, since a competitive market is assumed, and a l l individual investors are price takers, this first best solution is not feasible. Instead, w i t h the help of a market maker a second best solution might be achieved. E q u i l i b r i u m results Figures 4, 5, and 6 show the numerical results. A l l of the tables are tabulated i n the Edgeworth trade box format. T h i s means that the numerical results i n a specific box are the equilibrium results (after trade) for i n i t i a l endowments (before trade) that correspond to that specific box. Since the trade box is symmetric we provide only the numerical results for i n i t i a l endowments beneath the diagonal. Figure 4 shows the equilibrium price i n the limit order book structure and the market maker structure. One can see that i n the limit order book structure the equilibrium price is an increasing function of the distance from the diagonal. T h e initial endowments distance from the diagonal affect on price is very significant. T h e difference i n price between endowments on the diagonal (corresponding to P = 1.053) to endowments very far from the diagonal (corresponding to P = 1.698) is 61%. In the limit order book structure, volume is considered as exogenous and i f there is much risk sharing between the investors, there is more trade which results i n a much higher equilibrium price for the stock. Contrary to this, i n the market maker structure the price of the stock does not i n general depend on the distance from the d i a g o n a l .  11  W h a t is  important for the equilibrium price is the initial wealth distribution. One can observe that the price is higher when moving down and to the left. Thus, the wealthier are the buyers of the stock, and the poorer are the sellers of the stock, the higher is the equilibrium price of the stock. Figure 5 shows the equilibrium volume i n b o t h market structures. One can see that i n the limit order book structure volume depends solely on the distance from the diagonal. O n the contrary, i n the market maker structure, the market maker drives volume up i f investors are initially close to the diagonal. T h e closer investors are to the diagonal, the Note that the very marginal decrease in price that can be observed when going from a point on the diagonol downwards also corresponds to lower fees of the market maker due to the higher volume (see Figure 6). The actual transaction prices are P + e for a buyer of stock, and P — e for a seller of stock. 1 1  64  more the excess volume the market maker generates.  Thus, the t o t a l volume  generated i n the market maker structure is rather similar for the different i n i t i a l endowments. W e can also observe that the difference i n volume between the two structure decreases as we move away from the diagonal. Figure 6 shows how the fee decreases as we move away from the diagonal. T h i s is consistent w i t h the fact that the market maker has to exert less effort when the initial endowments of the investors produce a sufficient amount of volume anyway. A t the right b o t t o m corner of the trade box we have a region where the market maker participation constraint is not met. T h i s means that a market maker would not generate any excess volume to avoid having a negative profit. If the market maker charges fees and does not exert effort, then b o t h the volume of the market maker structure and the investors utilities are lower compared to the limit order book structure. Welfare comparison T h e question of whether a market maker structure is efficient or not comes down to a comparison of utilities. T h e results of this comparison are shown i n Figure 7. Note that the trade box is symmetric from b o t h sides of the diagonal. T h e figure shows that there are basically two regions for which a monopolist market maker structure Pareto Dominates the limit order book structure. T h e m a i n region is close to the diagonal where the volume generated i n a market w i t h a limit order book structure is small. Region 1 represents i n i t i a l endowments i n which investors of type 1 are better off w i t h a market maker structure, while investors of type 2 are better off w i t h a limit order book structure. Region 2 represents i n i t i a l endowments i n which investors of type 2 are better off w i t h a market maker structure, while investors of type 1 are better off w i t h a limit order book structure. Finally, the white region represents i n i t i a l endowments i n which the limit order book structure is the efficient trading structure.  In this  region, the investors would not have traded more w i t h a monopolist market maker t h a n without one. In this area there is much volume generated by the risk sharing criterion of simply moving to the diagonal and this reduces most of the future liquidity risk. T h e ability of the market maker to charge fees from the investors is very limited resulting i n the market maker's participation constraint not being satisfied. In order to get some economic intuition it is useful to t h i n k about the monopolist market maker effect on the market as made of three factors, which are not independent. Assume that the sellers are investor of type 1 (initial endowments below the diagonal), and we analyze the changes introduced by the market maker. 1. Fee charged effect- e(X\  — Xi)  - this fee affects the budget equation. 65  It  has a negative effect on the utility of b o t h investors. Note that the only a sum of c(V — Vo) is deadweight cost that could have been avoided by collusive behavior of the investors. T h e difference between the deadweight cost and the fees charged is the profit of the monopolist market maker. Thus, a central planner that could endogenize the volume consideration w i t h the same cost function as the monopolist market maker, could save the investors only part of the fees charged. 2. Trade revenue effect - ( P  m  — P ) (X\ —Xi), where P  m  is the equilibrium price  i n a market w i t h a market maker, and P the equilibrium price i n a limit order book structure. T h i s effect has a different sign for the two investors. When P  m  > P , investors of type 1 are better off because they sell the shares  for a higher price, while investors of type 2 are worse off because they buy the shares. T h e opposite is true when P  < P.  m  3.  Spread effect - the future liquidity of the stock is negatively correlated w i t h the equilibrium price. However, even i f P  = P , a market maker structure  m  future spread is still higher because of the higher volume generated, which is a function of e. B o t h investors hold some amount of shares after trade and are affected by this reduced spread. T h e effect is positive for b o t h investors. T h e important t h i n g to realize is that investors of type 1, being the sellers, always prefers a higher price for the stock. T h e y have a double positive effect on their utility: (a) a higher price of the stock represents a lower future spread, thus, a higher value for the shares they hold after trade, (b) they get more cash for the shares they sell. O n the other hand, investors of type 2 can be worse off because they b u y shares at a higher price. T h e curve P  m  = P (not drawn i n the figure)  represents endowments for which the price i n the limit order book structure and the the market maker structure is the same. T h i s curve would have divided the two small regions where a monopolist market structure is efficient. T h i s means that investors of type 1 can still be better off w i t h a market maker structure even if P  m  < P. T h e reason for that is the spread effect.  Before moving on w i t h an interpretation of the results, it is worthwhile to quantify the actual increase i n utility due to the market maker involvement i n the market. For this the Certainty equivalent (hence, CE) is calculated. Figure 8 panel a and b illustrate the CE for a fixed initial endowment of Xi = 0.9 and different cash endowments. It is calculated for b o t h types of investors. defined as the proportion of wealth reduction i n every s t a t e  12  CE is  that is needed i n  order to return to the same expected utility that existed i n an a limit order book Note that their are four states: A, A, B, B_.  66  market structure. If CE is negative t h a n the expected utility i n the limit order book structure is higher t h a n that of the market maker structure. P a n e l b , which zooms-in on a range of relevant cash endowments, shows that the lower Pareto Dominate region does not increase the welfare of the investors i n a substantial way.  T h e m a x i m u m increase of wealth i n that region is less t h a n 0.1%. T h e  result of this analyses suggests that the major improvement i n welfare is derived for the region close to the diagonal where there originally was not a high enough of volume generated by risk sharing alone. Welfare improvements due to the market maker are up to 11% for the sellers and up to 6% for the b u y e r s .  2.4  13  C o m p e t i t i o n between Market Makers T h e two trading mechanism analyzed i n the previous section are not the  only two types observed i n financial markets, but rather, they represent the two extreme possibilities on the array of possibilities between a monopolist market maker structure where a l l trades are done through a single market maker, and a competitive market where a l l trades are done directly between buyers and sellers. For example, the N Y S E specialist market is a monopolist market maker structure where every security listed is assigned a single specialist who is responsible for maintaining a fair and orderly market i n the security, the new Euronext exchanges i n E u r o p e operate by the limit order book structure w i t h no intermediary, while a partially competitive market maker structure such as N a s d a q lies somewhere i n the middle of these two extreme cases. I n this section we relax the assumption of the monopolist market maker and analyze how competition between market makers affects the equilibrium outcome. Cournot competition W e assume a Cournot competition where two or more market makers compete i n the market and each decides how much effort to exert. C o m p e t i t i o n on the volume generated, rather t h a n on price setting, is consistent w i t h the fact that financial markets have institutional features that i n general preclude price competition. These include the fixed m i n i m u m tick size that is large relative to the relevant prices, the existence of alternative methods (other t h a n price) for attracting order flow, and restrictions on spreads ( K a n d e l and M a r x (1997)). T h e widely publicized Christie and Schultz (1994) study about the price collusion of the market makers provide further justification for this view It is interesting to learn that the involvement of a market maker has a different effect depending on whether you buy or sell stock. However, if we assume that all investors in the market buy and sell stock at some point in time, this difference does not bear any real importance. 13  67  i n terms of the institutions of the Nasdaq market. I n our setting, the market makers decide on the amount of effort they wish to invest i n the search and promotion technology, given the competitors decisions. W e assume the following: A s s u m p t i o n 1: Search related costs are the same as i n the monopolist case. B y exerting effort and increasing volume, a market maker does not increase (or decrease) the search costs of other market makers. Thus, there is no economy of scale (or diseconomies of scale) i n having more t h a n one market maker. A s s u m p t i o n 2: T h e excess volume generated by the combined effort of a l l market makers is split uniformly between a l l market makers, no matter what each market maker actual effort i n generating this excess volume was. W e assume that by investing i n different search related costs and promotion schemes, a market maker can increase the t o t a l volume i n the market, however, this excess volume w i l l be shared w i t h a l l other market makers. Let m be the number of competing market makers.  T h e t o t a l amount of  excess volume generated i n the economy is represented by m  (11.19) where V is t o t a l excess volume, Vi is volume generated by market maker i, and e  Vo is the volume generated i n a competitive l i m i t order book structure.  We  represent the excess volume generated by market maker i by (11.20)  m  T h e excess volume Vi is denned as the extra volume over and above the p o r t i o n ^ of the volume that is generated i n a competitive limit order book structure. Definition  A Nash equilibrium is a set of excess volume (ti*, v ,.., v £ j satisfy2  ing Ki( l, 2, V  V  .., V*, .., V* ) > 7Ti(vl, V%, .., V .., V* ), V 1 m  U  m  P r o p o s i t i o n 2 In a symmetric  Cournot  generated is a decreasing function  of the number of market  Proof.  competition  the total excess volume makers.  G i v e n the same cost function of eq. (11.14), market maker k profit  function is, ^k(Vl,V ,..,V ,..,V ) 2  k  m  =  2 e ( £ ^ + Vo)  1  P  7 2~2 i  m  v  k  68  if  v v  > where 7 > 0,6 > 1  where the first t e r m on the R H S is the fee collected by the market m a k e r , while 14  the second t e r m is the cost of the market maker for generating the excess volume v . Note that the cost is distributed proportionally. Market maker k  proportion  k pays a  of the costs. T h e profit function can be simplified to, 7T {v ,V ,..,V ,..,V ) k  1  2  k  V (2_>i)  = —  m  k  r  7  m  Differentiate the profit function t o find the o p t i m a l v , k  dTT (v |P,£,1>1, . . , ^ - 1 , ^ + 1 , ..,v ) k  k  m  _  (6 -  dv  l)v ) k  k  In a symmetric equilibrium we have vi,v , ..,v , 2  ,v  k  m  = v  Thus, i n equilibrium 2ej m  P(m + S- 1)  and the t o t a l excess volume generated is  V = V-V e  0  2e7  = mv* =  i  6-1  P(m + 8- 1)  A s m increases the excess volume generated i n the market decreases. C o r o l l a r y 1 As m —• oo, there is no excess volume and the competitive maker structure equilibrium approaches the limit order book structure  • market  equilibrium.  V is a decreasing function of m and l i m (V ) = 0. Thus, as the number of e  e  m—too  market makers increases, i t cannot be a better alternative then the limit order book structure. I n b o t h cases, there is no excess volume. However, trade i n the market maker structure may bear some fee e, which makes investors worse off. If their is no excess volume i n the market maker structure, t h a n for any small e, the limit order book structure Pareto Dominates the market maker structure for a l l different i n i t i a l endowments. W e learn that once we assume some degree of competition between the market makers, the market makers have their own 14  IS  Note that volume divided by price gives us quantity.  (St, Vo) <+  69  Thus, the quantity traded in the market  free-rider problem. E a c h of the market makers prefers that the other market makers pay the search and promotion related costs that reduce future liquidity risk. In other words, a l l the market makers benefit from the fact that there is more volume i n the economy, while their cost is only dependent on their own excess volume generated. T h e result is that we have a free-rider problem from the market makers perspective. A s more market makers compete i n the market, each market maker can commit to a relatively lower total excess volume i n the economy. T h i s is translated to a relatively smaller reduction i n future liquidity risk. I n the limit, as the number of market makers becomes very large, no excess volume is generated and liquidity risk i n not reduced.  15  Since volume is a public  good, market makers cannot commit that they w i l l exert effort to generate more volume. W e have a double free-rider problem, which brings us back to no excess volume being generated similar to the Umit order book structure. The result of this subsection suggest that there are benefits to the monopolist market maker structure that are eliminated once there is some competition between market makers. W h i l e a monopolist market maker can endogenize the volume decision completely, a competitive structure cannot. In fact, a high degree of competition completely eliminates the ability to reduce liquidity risk and improve the welfare of investors. Numerical results Since any market maker can commit that i n aggregate excess volume w i l l be V , we have V = VQ + V . T h e resulting pricing schedule for the E  E  oligopoly market maker structure is e  =  P  (  m  - -v^  + 6  m  (  i  m  )  27  Solving numerically for the equilibrium is done similar to the monopolist case. The results, not surprisingly, show that volume always goes down as m increases. Since market makers can commit to less excess volume, the equilibrium t o t a l volume is lower. W e also learn that the fee e decreases w i t h the degree of competition m. Thus, as the market maker structure becomes more competitive it resembles the limit order book structure since b o t h excess volume and fee decrease. However, an interesting finding is that the price of the stock can either go up or down when m increases. It turns out that the price goes down when most of the wealth is endowed to the sellers, and the price goes up when most of the wealth is endowed to the buyers. In other words, competition between market Note that this result does not depend on the way we distribute the revenues from trade between the market makers. As long as this distribution is not only dependent on each of the market makers own effort, there will be some free riding between them, which in the limit has the same properties. 1 5  70  makers makes the price more informative about the actual demand and supply of investors. W h e n most of the wealth i n the economy is endowed to those who supply the stock, the price is lower t h a n i n the case when most of the wealth is endowed to those who demand the stock. T h e fact that competition between market makers may have different pricing affects, also means that competition has different welfare implication. If the oligopoly competition between market makers results i n a higher equilibrium price compared to the monopolist case, the revenue affect (as discussed i n section 2.3) makes investors of type 1 better off and investors of type 2 worse off, compared to the monopolist case. O n the other hand, i f the oligopoly competition between market makers results i n a lower equilibrium price compared to the monopolist case, t h a n the revenue affect makes investors of type 2 better off and investors of type 1 worse off compared to the monopolist case. Figure 9 presents the Edgeworth trade box for m = 1,2,4, and 10. In this figure we can observe how the different optimality areas change when m increases. I n general, the trade box is pressed from the two white regions inward towards the middle. A s the degree of competition increases, the area representing limit order book optimality increases (the white triangular areas). W e also see how the dark market maker optimality region around the diagonal becomes narrower. In the limit a l l the trade box would be white, and a very competitive market maker structure cannot be a better structure t h a n a limit order book structure.  2.5  Discussion O u r analyses shows that a monopolist market maker can increase welfare and  reduce the liquidity risk only i n markets where there is not a high need for risk sharing.  16  O n l y i n these markets, the resulting equilibrium trade volume is not  sufficient to reduce the liquidity risk. In these markets, a monopolist market maker that interacts w i t h the buyers and the sellers, may i n fact reduce this negative externality and by doing so increase the investors welfare. T h i s means, that market makers are important for liquidity purposes only i n economies where risk sharing needs would not have produced the necessary volume needed to reduce the liquidity risk. Thus, our welfare analysis suggests that the monopolist market maker structure is optimal i n economies where the opportunity for improving risk sharingjs small. As shown in the previous section, this is true to a lesser extent in markets where there is some degree competition between market makers. 1 6  71  O p p o r t u n i t y for improving risk sharing In order to calculate the opportunity for improving risk sharing for different economies, we refer back to Figure 7. L e t the two types of investors represent two groups of investors who differ the most i n terms of their endowments. A natural way of dividing investors i n the economy i n such a way, is to divide the investors to controlling shareholders (hence,  CS)  and ordinary investors (hence, OS). T h e CS investors hold a substantial percent of their wealth i n the equity of a particular company and their future wealth depends to a large extent on the future value of that company's equity.  17  The  OS investors on the other hand, hold very diversified portfolios and have no particular stake i n any one company. I n terms of the model, assume investors of type 1 are i n the CS group, and that investor of type 2 are i n the OS group. A point i n the trade box represents a certain endowment distribution of equity and cash that is divided between the CS and the OS groups. Depending on where that point is located, we can evaluate whether a market maker structure is efficient or not.  F r o m Figure 7  we learn that when the amount of cash and equity holdings is similar (close to the diagonal), then the market maker structure is efficient and a rationale for a market maker structure exists. O n the other hand, when one of the groups controls most of the equity and the other controls most of the cash, the market maker structure does not Pareto Dominate the limit order book structure. Due to the high opportunity for improving risk sharing between the two groups, high volume is going to be generated anyway. Thus, uncertainty is reduced without the need of the extra liquidity that a monopolist market maker structure can provide. A monopolist market maker structure is o p t i m a l when the distribution of wealth between the two groups is close to the diagonal of the Edgeworth box, while a limit order book structure is o p t i m a l when the distribution of wealth between the two groups is far from the diagonal. We know that the CS investors are endowed w i t h relatively more equity t h a n cash. Otherwise, our definition of controlling shareholders w i l l be inconsistent w i t h the idea that they are investors who hold a relative large amount of equity. W e let the CS investors be investors of type 1 i n Figure 7. W e can see that as long as the CS hold i n aggregation less t h a n 30% of the total amount of equity, a market maker structure is efficient. A s the CS group gets larger, one needs to know C S ' s non-equity holding i n order to speculate on the efficient market structure.  Note however, that as the CS  investors holding of equity grows, its actual holding of cash decreases. T h i s has to be the case, unless we assume that one group controls almost a l l of the wealth Note that although Bill Gates is a wealthy individual with many different assets, his total wealth depends to a large extent on the value of Microsoft's shares. In other words, he is not well diversified. 17  72  (including labor payments) i n the economy. T h i s formulation allows us to arrive at the following empirical hypothesis: A monopolist market maker structure is efficient for shares that are mostly held by ordinary shareholders, while the limit order book structure is justified for shares that have a higher percentage holding by the controlling shareholders. In other words, i f controlling shareholders hold a h i g h percentage of the equity i n an economy, there is high opportunity for risk sharing. Therefore, a market maker structure is not efficient.  18  T h i s implies  that shares traded on the N Y S E should be mostly held by ordinary investors who invest i n shares either directly or through institutional investors for diversification reasons. Conversely, i n the limit order book structure, we would expect to see shares w i t h a high percentage of holding by the controlling shareholders.  One  can also t h i n k about this interpretation i n terms of a life-cycle of a public firm. We would expect more cases of initial public offering i n a competitive limit order book structure market. W i t h time, as the controlling shareholders hold less of the t o t a l amount of shares, excess volume generating becomes more important, so companies may wish to be traded on a partially competitive market maker structure such as Nasdaq.  Finally, when the company is mature and widely  held, the monopolist market maker structure becomes desirable which can lead companies to move to exchanges such as the N Y S E .  1 9  P o l i c y implications T h e study shows that inefficiencies are created i n a competitive environment. T h i s occurs because investors assume away the fact that their own trade affects market liquidity. H a v i n g a market maker which sets the b i d ask quotes can solve this free-rider problem. However, this is true as long as the market maker is not subject to a free-rider problem due to the competitive environment i n which it acts. A monopolist market maker is able to set a pricing schedule that relates the individual investor's own volume to the fee it pays. A t the same time, investors understand that the same pricing schedule is given to Note that since this is a one period model, it is not clear how one can distinguish between the investors after they trade. In other words, how do we think of this problem in a fully inter-temporal setting. One way to approach the issue is to focus on the essentials: in an inter-temporal setting new equity is being issued each period and their are both liquidity and endowments shock, so even after trade we can expect that there is still some opportunity for improving risk sharing. We would argue that if initially the economy was far from the diagonal, then after trade it would still be rather far from the diagonal. 18  We presented in this subsection only one way of measuring the opportunity for improving risk sharing in the economy. There are other ways for measuring this variable. For example, estimating the percentage of the population who hold directly or indirectly (through mutual funds and pension funds) stocks of domestic companies of a country. A high percentage of the population that holds stock is a measure that tells us that the economy is developed in the sense that their has been much risk sharing and there is less opportunity for improving risk sharing compared to countries where only a low percentage of the population holds stocks. 1 9  73  all investors i n the economy, and the result is that they take into account their own trade effect on liquidity. The policy implication for the regulators is that there are benefits to a monopolist structure compared to the competitive structure. In broad terms, if one agrees that liquidity has the property of a public good, then a non-competitive trading environment can reduce this negative externality. T h i s means that the N Y S E specialist structure might be efficient i f liquidity is an important issue. A s for more competitive environments of market makers such as Nasdaq, it is important to realize, that even though it is commonly stressed that a competitive environment reduces the bid-ask spreads, it may also reduce the liquidity i n the market. T h i s is important i n light of the S E C decision that the N A S D should develop an alternative display facility to provide an O T C market that w i l l be able to compete w i t h N a s d a q .  20  T h e major motivation for the proposed change  is to increase competition between the different designated dealers as shown by IM-2110-5. ( A n t i - I n t i m i d a t i o n / C o o r d i n a t i o n ) : "The B o a r d of Governors is issuing this interpretation to codify a longstanding policy. It is conduct inconsistent w i t h just and equitable principles of trade for any member or person associated w i t h a member to coordinate the prices (including quotations), trades, or trade reports of such member w i t h any other member or person associated w i t h a member; to direct or request another member to alter a price (including a quotation); or to engage, directly or indirectly, i n any conduct that threatens, harasses, coerces, intimidates, or otherwise attempts improperly to influence another member or person associated w i t h a member  "  T h e increased competition agenda should be considered i n light of the negative externality that are discussed i n the previous sections. It is also not clear why a competitive market maker structure is desired when trading structures such as the limit order book structure are feasible today at very low cost. If the analysis i n this study is to be believed the regulators should at very m i n i m u m be aware that higher competition may i n fact reduce the liquidity i n the market. E m p i r i c a l Evidence O u r theory suggests that market makers should exist i n countries where there is less opportunity for improving risk sharing. T o check our theory we surveyed the major equity exchanges i n different countries. T h e survey was conducted for countries that pass the following two criteria (i) have a On December 14, 2001, the SEC came out with a proposed rule change relating to Nasdaq's proposed separation from the NASD and the establishment of the NASD Alternative Display Facility (ADF). The A D F is a facility that should provide memebers with the ability to do OTC trading in listed securities once Nasdaq is registered as a for-profit exchange. It would allow market participants to send their quotations without having to go through Nasdaq 2 0  74  G D P per capita of $12,000 or more, (ii) have a G D P which is higher t h a n 0.15% of the world G D P . These two selection criteria leave us w i t h the richest countries i n the world but excludes a number of them that do not have a significant stock market for domestic companies. W e are left w i t h 27 countries.  21  T h e survey had  two questions for the two pieces of data information needed to test the validity of our model: (a) a question on whether trade must go through a market maker (b) a question on the opportunity for improving risk sharing for domestic companies. Unfortunately, for many countries the answer for (b) was unavailable. It also turns out that only i n the U n i t e d States and the U n i t e d K i n g d o m investors trade i n domestic equity is facilitated w i t h market makers that provide bid-ask quotes.  22  Thus, we cannot provide solid proof to our theory. T h e only t h i n g we  can say is that the investors i n b o t h the U n i t e d States and U n i t e d K i n g d o m are more diversified t h a n the investors i n a typical country i n our survey and that is consistent w i t h our theory.  2.6  Concluding Remarks There are a few empirical studies that are consistent w i t h the notion that  the regulators decision on the type of exchange structure matters. These studies show that the choice of exchange structure can influence b o t h the pricing and the volume of the securities t r a d e d .  23  T h e N Y S E points to the "specialist system"  as a superior form of market organization, citing the "increased liquidity" i n its promotional literature. Similar promotions can be find i n other market maker exchanges, such as Nasdaq. O n the other hand, it is rather hard to find an exchange w i t h a limit order book trading structure promoting its services due to its increased liquidity. Compatible w i t h this notion, the theory developed i n this study suggests that a monopolist market maker structure can increase liquidity of a stock compared to a competitive limit order book market structure. A monopolist market maker can solve a free-rider problem that occurs when individual investors do not take into account their own trade effect on market liquidity. Since the US equity market has two major exchanges, the NYSE and NASDAQ, we conduct our survey for 28 exchanges. 2 1  Note, however, that the total volume traded in the US and U K was approximately 73% of the total volume traded in these 27 countries equity markets in the year 2000. 22  Mccorry, and Frino (1995) compare the costs of trading on NYSE, a specialist market, with the Australian Stock Exchange which operates an electronic open limit order book. The results suggest that the electronic open limit order book provides lower execution costs after controlling for stock price, trade size, trading activity and price volatility. Corwin (1999) and Neal (1992) provide additional evidence that market exchange structure matters. 23  75  W h e n allowing for market makers competition, we learn that the market makers have their o w n free-rider problem. E a c h of the market makers prefers that the other market makers pay the search and promotion related costs that reduce future liquidity risk. Thus, while a monopolist market maker can endogenize the volume decision completely, a competitive structure cannot. W h e n comparing the monopolist market maker structure and the limit order book structure it is found that the initial endowments of the investors play an important role. Depending on the opportunity for risk sharing that is present i n the economy, a monopolist market maker structure can be efficient. B y dividing the economy into two group of investors representing the ordinary shareholders and the controlling shareholders, we are able derive some empirical implications concerning the optimality of the market structure.  76  REFERENCES CITED  1. A . A d m a t i and P. Pfleiderer (1988), " A Theory of Intraday Patterns: Volume and Price Variability." T h e Review of F i n a n c i a l Studies 1, 3-40. 2. Y . A m i h u d , and H . Mendelson (1986),  "Asset P r i c i n g and the  Bid-Ask  Spread." Journal of F i n a n c i a l Economics, 223-249. 3. Y . A m i h u d , and H . Mendelson, and B . Lauterback (1997), "Market M i crostructure and Securities Values: Evidence from the T e l A v i v Stock E x change." Journal of F i n a n c i a l Economics 45, 365-390. 4. B . Biais, L . Glosten, and C . Spatt (2000), "Market Microstructure." W o r k i n g paper, Carnegie M e l l o n University. 5. W . Breen, L . Hodrick, and R . K o r a j c z y k (1999), "The Determinants of E q u i t y Illiquidity." W o r k i n g paper, Northwestern University. 6.  S. Brusco and M . Jackson (1999), "The O p t i m a l Design of a Market." Journal of Economic Theory 99, 1-39.  7.  S. Chakravarty and A . Sarkar (1999), "Liquidity i n U . S . F i x e d Income M a r kets: A comparison of the B i d A s k Spread i n Corporate, Government  and  M u n i c i p a l B o n d Markets." W o r k i n g paper N o . 73, Federal Reserve B a n k of New York. 8. W . Christie and P . Schultz (1994), " W h y D o M a r k Makers A v o i d O d d - E i g h t h Quotes?" Journal of Finance 49, 1813-1841. 9.  G . Constantinides  (1986), " C a p i t a l Market E q u i l i b r i u m w i t h Transaction  Costs." Journal of Political Economy 94, 842-862. 10. M . Coppejans, I. Domowitz, and A . M a d h a v a n (2001), " L i q u i d i t y i n an Automated Auction.", S S R N F E N . 11.  S. C o r w i n (1999), "Differences i n Trading Behavior across N Y S E Specialist  F i r m s . " Journal of Finance 54, 721-745. 12. A . D i x i t (2001), " O n Modes of Economic Governance.", C E S i f o W o r k i n g paper N o . 589 13. D . Duffle, N . Garleanu, and L . Pedersen (2000), "Valuation i n D y n a m i c Bargaining Markets." W o r k i n g paper, Stanford University. 77  14. N . Economides and A . 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Pagano (1989), "Trading Volume and Asset Liquidity." Quarterly Journal  of Economics, 255-274. 26. A . R u b i n and L . G a r l a p p i (2001), "Investors' Preference and L i q u i d i t y Needs i n the P r i c i n g of Corporate Debt." W o r k i n g paper, U B C . 78  27. A . Sarin, K . Shastri, and K . Shastri (1996), "Ownership Structure and Stock M a r k e t Liquidity." S S R N F E N . 28. D . Scharfstein and J . Stein (1990), "Herd Behavior and Investment." A m e r ican Economic Review 80, 465-479. 29.  M . Spiegel and A . Subrahmanyam (1995), " O n Intraday R i s k P r e m i a . " Jour-  n a l of Finance 50, 319-340. 30. D . Vayanos (1998), "Transaction Costs and Asset Prices: A D y n a m i c E q u i l i b r i u m M o d e l . " Review of F i n a n c i a l Studies 11, 1-58. 31.  D . Vayanos and J . V i l a (1998), " E q u i l i b r i u m Interest R a t e and L i q u i d i t y  P r e m i u m w i t h Transaction Costs." W o r k i n g paper, M I T .  79  A p p e n d i x A For a proof of the feedback loop between liquidity and volume, we follow the formulation of an endogenous liquidity shock as i n R u b i n and G a r l a p p i (2001). Let us consider a two-period model i n which n identical homogeneous investors choose a portfolio holding at time t = 0. T h e portfolio holdings are made of cash (co), earning no interest, and of one type of risky security whose supply per investor is x . T h i s means that the total supply of the risky security i n the 0  market is nx . 0  T h e security is liquidated at t = 2 and its payoff is k n o w n to  be either A or B w i t h probability of (1 — TX) and TX respectively. A t t = 1, a portion p of the investors receive a negative liquidity shock and are forced to sell all their holdings to the portion of investors who are not affected by the liquidity shock. A t t = 0, the investors do not know whether they w i l l be affected by the liquidity shock or not, however, after the event occurs, the two groups of investors act according to their new consumption needs. T h e group of investors who have the liquidity shock sell a l l their security holdings and consume, while those who continue to t = 2, update their portfolio holdings and purchase the new supply of securities. Thus, at t — 1, some of the investors may have high consumption needs which forces them to sell a l l their holdings. Those who are not hit by this liquidity shock, buy these shares i n the market, and consume only at t = 2, when the payoff of the securities is distributed. B y the assumption of homogeneity of the investors, ex-ante each investor assumes w i t h probability p that he may experience a liquidity shock. A t t = 0 the homogeneous investors are endowed w i t h an initial wealth of WQ which is used to buy the risky security. T h e remainder of wealth is invested i n cash, thus,  WQ  =  PQXQ  + c$. A t t = 1,  the investors are split into two groups, those who have the liquidity shock and sell their shares i n the market and those who buy these shares i n the market and consume only at t — 2. Trades at t = 1 are performed at the equilibrium price P i . Note that the portfolio decision of t = 1 depends b o t h on the degree of the liquidity shock and the portfolio composition at t = 0. For example, when the liquidity event is p there is a supply of pxQ shares to the market at t = 1 that must be absorbed by a p o r t i o n of (1 — p) of the investors, who constitute the fraction of investors who continue to time t = 2. Since these investors are homogeneous as well, the new supply of securities per investor  24  at time t = 1  equals XQP  W e assume the existence of individual risk averse investors whose preferences The new supply of securities in the market is nxop . t  80  over wealth w are described by a concave utility function. Investors maximize terminal wealth, pU{P  + (w - P x) + (1 - p)(l - TT)U (fi  lX  0  + w -  0  Px -  0  tef)  0  max XQ  (Al) However, note that all uncertainty is known at t = 0, meaning that P = Pi. This simplifies the problem to, 0  max  + (1 - p)(l -  U(w )  P  0  (jfZJ  + °)  x  TT)U  XQ  + ( ~  w  x  P)*  { T ^ J  U  P  X  P  +  )  WO  (A2)  which results in the following first order condition, (l-Tr)tf'  (YZJ  X  +  ) ( -  WO  A  p  o)+7rc/'  {^rzy  + ™o)  x  (B -  P) 0  =  o (A3)  For the equilibrium to be free of arbitrage it must be that B < PQ < A. A n increase in p results in an increase in but a reduction in ~ >. This means B  r  l-p  p<  1-p  that the L H S of the equation is reduced in value (U" < 0) , but also the R H S is reduced in value because it is multiplied by a negative quantity of B — PQ. Thus, in order to satisfy the first order condition, Po must go down. A higher liquidity shock p at t = 1, reduces the price Po of the risky security at t = 0. Since x is fixed, the dollar volume of traded stock at the t = 0 is reduced as well. Appendix B The parameters of the economy are assigned the following values:  0=  a = 0.2,  0.15, X = 1, C = 1  We also assume that investors have a log utility function and that they assign equal probabilities for the up and down states, i.e., TT = 0.5. After deriving the equilibrium variables (P, X\, X , e) we compare the expected utilities between the market structures. Following is a description of the equilibrium derivation in both market structures. 2  Limit  Order Book  Structure:  The two F O C for the limit order book structure are: A +  C + PX i  i  ~aT77 — P  p+v + X (A i  -  1  + -fy-P)  +. =  =  d  + PXi  + =  =  B + ^ - - P P + V  d  + PXi  + XiiB  + ^y-P)  A  ^p+v  + X i i A - ^ y - P )  (B  B - ^ - P d  + PXi  81  ^  + X i i B - ^ y - P )  = 0, for i = 1,  and the clearing conditions is  X + X = 1 x  2  In equilibrium V = Vo, where Vo is the conjecture that investors have o n volume during trade.  Solving for the three unknowns Xi, X , a n d P is done b y a n 2  iterative process as described i n section 3.2.  Market Maker Structure: W e set the market maker cost function to be quadratic (6 = 2) w i t h 7 = 15.Since volume becomes an endogenous variable, the two F O C of the investors are:  2P  2  aPXi  PVo  l y  X* — Xi  7  [p+P\Xi-X \]  Ci + PXi-^V-Vo) 2P  2  Xi-Xi  yv,' — -V A ? I —EVa — +  \~Y  d + PXi-^V-Vo) 2P  2  ! Xi — Xi  ?  Ci + PXi-Z(V-V ) 0  2P  1  2  IXv i - X i \ - ^  d + PXi-^V-Vo)  +  ( \  +  P  +  {B  +  V  1  (B2)  ^y-P) T=^— [p+P\Xi-Xi  +Xi(A-^y-P) £  Xi-Xi  V  - P )  +  +Xi(B  ( B - ^ - P ) Xi-Xi  +  B^V ~ ) + 7 P )  ~  A  +Xi(A  Xi-Xi  E  j  t  +  aPXj [p+P\Xi-Xi\]  + p+v ^v-P) aPXj  [p+p\Xi-Xi\]  = 0  + Xi{B - tffc - P)  For i = 1,2, where V = P \ Xi — Xi | , and Vo is the equilibrium volume of the limit order book structure. F r o m the two F O C conditions, the clearing condition and the market maker schedule (eq. 11.13) one can derive e, P, X\, 82  X. 2  Figure II.1: A schematic illustration of the feedback loop between volume and liquidity: VQ— volume at time t = 0; A, B — time t = 1 expected price of the stock; a A, C B ~ variance of good and bad state prices at t = 1.  83  1  0.9  0.8  0.7  0.6  0.5  0.4  0.3  0.2  0.1  Figure II.2: The Edgeworth Trade Box and the formulation of volume and price depending on the initial endowments of the investors.  84  0  Figure II.3: A schematic illustration of the derivation of the market maker pricing schedule: S(Pi)— the supply curve of excess volume that the market maker provides the market for a given Pf, MR(Pi)—  the marginal revenues curves derived from investors demand for excess  volume for a given price Pi.  Price in Order Book Structure 0.950 0.900 0.850 0.800 0.750 0.700 0.650 0.600 0.550 0.500 0.450 0.400 0.350 0.300 0.250 0.200 0.150 0.100 0.050  1.053  1.053  1.053  1.249  1.053  1.249  1.378  1.053  1.249  1.378  1.464  1.053  1.249  1.378  1.464  1.522  1.053  1.249  1.378  1.464  1.522  1.563  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.646  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.646  1.657  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.646  1.657  1.666  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.646  1.657  1.666  1.673  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.646  1.657  1.666  1.673  1.679  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.646  1.657  1.666  1.673  1.679  1.684  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.646  1.657  1.666  1.673  1.679  1.684  1.688  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.646  1.657  1.666  1.673  1.679  1.684  1.688  1.692  1.053  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.646  1.657  1.666  1.673  1.679  1.684  1.688  1.692  1.695  1.249  1.378  1.464  1.522  1.563  1.593  1.615  1.632  1.646  1.657  1.666  1.673  1.679  1.684  1.688  1.692  1.695  1.698  0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950  Price in Market Maker Structure 0.950 0.900 0.850 0.800 0.750 0.700 0.650 0.600 0.550 0.500 0.450 0.400 0.350 0.300 0.250 0.200 0.150 0.100 0.050  1.452  1.952  1.599  1.538  1.644  1.616  1.579  1.669  1.649  1.627  1.601  1.686  1.670  1.653  1.636  1.616  1.701  1.686  1.671  1.657  1.642  1.627  1.714  1.700  1.686  1.672  1.660  1.647  1.635  1.727  1.712  1.699  1.686  1.674  1.662  1.652  1.642  1.739  1.724  1.711  1.698  1.686  1.675  1.665  1.655  1.647  1.751  1.736  1.722  1.709  1.697  1.686  1.676  1.667  1.659  1.652  1.763  1.748  1.733  1.720  1.708  1.697  1.686  1.677  1.669  1.662  1.656  1.775  1.759  1.745  1.731  1.718  1.707  1.696  1.687  1.678  1.671  1.665  1.659  1.787  1.771  1.756  1.742  1.729  1.717  1.706  1.696  1.687  1.680  1.673  1.667  1.663  1.801  1.783  1.768  1.753  1.739  1.727  1.715  1.705  1.696  1.688  1.681  1.675  1.670  1.665  1.815  1.797  1.780  1.764  1.750  1.737  1.725  1.714  1.704  1.696  1.688  1.682  1.676  1.672  1.668  1.832  1.811  1.793  1.776  1.761  1.747  1.734  1.723  1.713  1.704  1.696  1.689  1.683  1.678  1.674  NA  1.852  1.829  1.808  1.789  1.772  1.758  1.744  1.732  1.721  1.712  1.703  1.696  1.689  1.684  1.679  NA  NA  1.883  1.851  1.826  1.804  1.785  1.769  1.754  1.741  1.730  1.720  1.711  1.703  1.696  1.690  1.685  NA  NA  NA  1.888  1.851  1.823  1.801  1.782  1.766  1.751  1.739  1.728  1.718  1.710  1.702  1.696  1.690  NA  NA  NA  NA  0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950  X 1 ->  Figure II.4: T h e equilibrium price of the risky asset i n the limit order book structure and 86 the market maker structure.  V I, -  Volume in Order Book Structure  <- X  2  0.950  0.000  0.900  0.000 0.028  0.850  0.000 0.028 0.058  0.800  0.000 0.028 0.058 0.089  0.750 0.700  0.000 0.028 0.058 0.089 0.121 0.000 0.028 0.058 0.089 0.121 0.152  0.650  0.000 0.028 0.058 0.089 0.121 0.152 0.184  0.600  0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216  0.550  0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248  0.500  0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248 0.280  0.450  0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248 0.280 0.312  0.400  0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248 0.280 0.312 0.344  0.350  0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248 0.280 0.312 0.344 0.376 0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248 0.280 0.312 0.344 0.376 0.407 0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248 0.280 0.312 0.344 0.376 0.407 0.439  0.300 0.250 0.200  0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248 0.280 0.312 0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248 0.280 0.312 0.344 0.100 0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248 0.280 0.312 0.344 0.376 0.050 0.000 0.028 0.058 0.089 0.121 0.152 0.184 0.216 0.248 0.280 0.312 0.344 0.376 0.407 0.150  c  2  i  0.344 0.376 0.407 0.439 0.471 0.376 0.407 0.439 0.471 0.503 0.407 0.439 0.471 0.503 0.535 0.439 0.471 0.503 0.535 0.566  0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950  Volume - Market Maker Structure <r~  X  2  0.950  0.239  0.900  0.305 0.282  0.850  0.329 0.322 0.309  0.800  0.340 0.338 0.335 0.329  0.750  0.347 0.347 0.347 0.346 0.344  0.700  0.351 0.352 0.354 0.356 0.357 0.358  0.650  0.353 0.355 0.358 0.361 0.364 0.367 0.370  0.600  0.355 0.358 0.361 0.365 0.369 0.373 0.377 0.382  0.550  0.356 0.359 0.363 0.367 0.372 0.377 0.382 0.387 0.393  0.500  0.357 0.360 0.364 0.369 0.374 0.379 0.385 0.391 0.397 0.404  0.450  0.357 0.361 0.365 0.370 0.375 0.381 0.387 0.393 0.400 0.407 0.415  0.400  0.357 0.361 0.365 0.370 0.376 0.382 0.388 0.395 0.402 0.409 0.417 0.426  0.350  0.356 0.360 0.365 0.370 0.376 0.382 0.389 0.396 0.403 0.411 0.419 0.427 0.437  0.300  0.355 0.359 0.364 0.370 0.376 0.382 0.389 0.396 0.404 0.411 0.420 0.428 0.438 0.447  0.250  0.353 0.358 0.363 0.369 0.376 0.382 0.389 0.396 0.404 0.412 0.420 0.429 0.438 0.448 0.458  0.200 0.150 0.100  0.350 0.355 0.361 0.368 0.375 0.381 0.389 0.396 0.404 0.412 0.421 0.430 0.439 0.448 0.458 0.344 0.351 0.358 0.365 0.373 0.380 0.388 0.396 0.404 0.412 0.421 0.430 0.439 0.449 0.459 0.334 0.344 0.353 0.362 0.370 0.378 0.386 0.395 0.403 0.412 0.421 0.430 0.439 0.449 0.459 NA  0.050 0.309 0.331 0.345 0.356 0.366 0.375 0.384 0.393 0.402 0.411 0.420 0.430 0.439 0.449 0.459  NA  NA  NA  NA  NA  NA  NA  NA  NA  0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950  Xj ->  Figure II. 5 : Volume in the limit order book strucutre and the market maker structure.  87  c i  2  £  —  Market Maker Fee  0.950  0.023  0.900  0.033 0.026  0.850  0.036 0.032 0.026  0.800  0.038 0.034 0.030 0.026  0.750  0.039 0.036 0.032 0.028 0.024  0.700  0.040 0.036 0.033 0.029 0.026 0.022  0.650  0.040 0.037 0.034 0.030 0.027 0.024 0.020  0.600  0.041 0.038 0.034 0.031 0.028 0.024 0.021 0.018  0.550  0.041 0.038 0.035 0.031 0.028 0.025 0.022 0.019 0.016  0.500  0.042 0.038 0.035 0.032 0.029 0.026 0.022 0.019 0.017 0.014 0.042 0.039 0.035 0.032 0.029 0.026 0.023 0.020 0.017 0.014 0.011  0.450  0.042 0.039 0.036 0.032 0.029 0.026 0.023 0.020 0.017 0.014 0.012 0.009 0.042 0.039 0.036 0.033 0.029 0.026 0.023 0.020 0.017 0.015 0.012 0.009 0.007 0.043 0.039 0.036 0.033 0.030 0.027 0.023 0.020 0.018 0.015 0.012 0.009 0.007 0.004  0.400 0.350 0.300  0.043 0.039 0.036 0.033 0.030 0.027 0.024 0.021 0.018 0.015 0.012 0.010 0.007 0.005 0.002 0.043 0.040 0.036 0.033 0.030 0.027 0.024 0.021 0.018 0.015 0.012 0.010 0.007 0.005 0.002 NA  0.250 0.200  0.042 0.039 0.036 0.033 0.030 0.027 0.024 0.021 0.018 0.015 0.012 0.010 0.007 0.005 0.002  0.150  0.050  NA  NA  NA  NA  NA  NA  NA  NA  NA  0.800  0.850  0.900  0.950  0.042 0.039 0.036 0.033 0.030 0.027 0.024 0.021 0.018 0.015 0.012 0.010 0.007 0.005 0.002  0.100  0.040 0.038 0.035 0.032 0.029 0.026 0.024 0.021 0.018 0.015 0.012 0.010 0.007 0.005 0.002 0.050  0.100 0.150 0.200  0.250  0.300  0.350  0.400  0.450  0.500  0.550  0.600  0.650  Figure II.6: Fee i n the market maker structure.  88  0.700  0.750  1  0.9  0.8  0.7  0.6  0.5  0.4  Market Maker Structure Optimal Investor 1 is Better off with Market Maker Structure, Investor 2 is Better off with Limit Order Book Structure. Investor 2 is Better off with Market Maker Structure, Investor 1 is Better off with Limit Order Book Structure. Limit Order Book Structure Optimal  Figure II. 7: Welfare comparison between the limit order book trade structure and the market maker trade structure.  89  Panel a  12  2I 0.25  i  i  i  ^  0.3  0.35  0.4  0.45  gure II.8: T h e Certainty Equivalent (CE)  for initial stock endowments of Xi  2 = 0.1 and different cash endowments (x-axis), where C  2  90  = 1 — C\.  Figure II.9: Welfare comparison between the limit order book trade and the market maker structure for different degrees of competition among market makers, m is the number of market makers. (Black region - market maker structure optimal. W h i t e region - limit order book structure optimal). 91  C H A P T E R III T H E EFFECT OF SHORT SALE CONSTRAINT R E M O V A L O N VOLATILITY IN T H EPRESENCE OF HETEROGENEOUS BELIEFS  3.1  Introduction T h e Black & Scholes (1973) model for option pricing relies on the assumption  that trading i n the option has no effect on the underlying asset. B l a c k & Scholes assume complete markets i n which the traded underlying securities span the distribution of final payoffs. In such a market options are redundant securities and do not affect the underlying asset.  However, we know that markets are  not always complete and options are not always redundant.  For example, the  emerging markets that are now engaging i n adapting new derivatives are probably much less then complete. Regulators often seem concerned about the effect that options have on the underlying asset.  1  T h i s clearly means that they i m p l i c i t l y  reject the assumption of completeness. T h i s chapter examines the incompleteness that arises from short sale constraints.  W e study how the elimination of short sale constraints affects the  volatility of the underlying asset. In many emerging markets the ability to short sell is nonexistent. E v e n i n developed markets such as the U . S . , many investors face constraints when t r y i n g to short sale, (e.g., A s q u i t h &; Meulbroek, 1995, Danielsen & Sorescu, 1999). O n the other hand, the ability to trade derivatives effectively removes short sale constraints and allows investors to bet against the market. Therefore, understanding of short sale constraints enhances our knowledge to an important aspect that derivative trading have on financial markets. T h e chapter is aimed at readers who appreciate a simplified approach for understanding the change i n volatility. W e examine a market that contains as many assets as future states but which is incomplete i n the absence of short sales. T h e intuition is derived from simple geometry: W e show that the price function of the risky asset i n a constrained economy is a concave function of the uncertain future payoff. W h e n constraints are eliminated, the price function becomes less concave (closer to linear). T h e reduction i n concavity allows us to derive predictions concerning the change i n volatility. W e show that volatility can increase or decrease depending on the variability of news and the other exogenous variables.  A l t h o u g h elimination of constraints provide benefits to the market  (i.e., increased welfare), they may reduce price stability. However, we show that 1  "The Economist", November 27,1999, pages 81-82. 92  volatility can also decrease when moving to an unconstrained economy. Thus, even though negative information can be more effectively revealed when short sale constraints are introduced, this does not mean that volatility w i l l necessarily increase. Our results are derived from a repetition of a static framework model, where investors have different beliefs about the future state of the economy, m a k i n g some of the investors want to short sale. A dynamic framework (e.g., HoUineld and Gallmeyer, 2002) has the undesirable property that i n the long r u n investors w i l l be homogenous and posses the same beliefs. T h a t means that i n the long run there should be no volatility and any non-zero volatility measure does not represent a "steady state" economy. O n the other hand, the idea that people w i l l have different beliefs also i n the long r u n seems realistic to us. If one agrees w i t h this assessment, a static framework is sufficient to derive results on volatility. A s an empirical illustration, we study the effect of index option initiation i n Israel.  Options were introduced i n Israel at a time when short sales were  prohibited, presenting an opportunity to examine the removal of a short sale constraint. W e find that option introduction increase was accompanied by a n increase i n volatility of the underlying index.  T h i s is of interest particularly  because previous empirical results have found that introduction of options was accompanied by either reduced return volatility of the underlying asset or no change i n volatility.  2  Our approach is related to two strands of the economics and finance literature.  One is the literature that deals w i t h short sale constraints starting w i t h  M i l l e r (1977), who argues that i n a market w i t h short sale restrictions, the price of risky assets is higher t h a n i n a market w i t h no restrictions. Similar, i n our model, the ability to short sell increases the supply of shares i n the market by the amount of the outstanding short position. It moves the vertical supply curve, reduces the price, and allows negative opinions to affect the market value of the stock. Jarrow (1980) examines the influence of a short sale restriction on the prices of risky assets i n a single period model w i t h two assets. H e demonstrates that, under heterogeneous beliefs, a short sale constraint can cause a decline i n the price of one risky asset and a rise i n the price of the. other risky asset due to a substitution effect. D i a m o n d and Verrecchia (1987) conclude that relaxing the short sale constraint increases the speed of adjustment to private information, especially to b a d news. T h e y also claim that short selling makes excess returns smaller i n absolute value and makes the distribution of excess returns on public For example: Bollen (1998), Chatrath et al (1995), Chaudhury and Elfakhani (1997), Damodaran and Lim.(1991), Detemple and Jorion (1990), Long et al (1994), St. Pierre (1998). 2  93  information announcement days less skewed to the left. These studies do not deal w i t h volatility changes of the underlying asset. Second, there exist a considerable number of theoretical papers that study the effect of financial innovation i n general. O u r study is related to a subset of this literature that considers the effect of additional derivative securities on the informational and allocational efficiency of the market (Grossman, 1988, K r a u s and S m i t h , 1996, H u a n g and W a n g , 1997, Cao, 1999). However, none of these studies focuses on the informational effect of relaxing the short sale constraint. Thus, we differentiate our analysis from these two strands of literature by studying the particular case of short sale constraints and volatility effects. T h e rest of the study proceeds as follows. In section 3.2 the model is developed and the differences between a constrained economy and an unconstrained economy are formulated. In section 3.3 we study how price volatility changes u p o n removal of the short sale constraint. O u r empirical application is i n section 3.4. Finally, i n section 3.5 we summarize and conclude.  3.2  The Model T h e specific assumptions defining the investors and the capital markets are  chosen i n the simplest setting possible: 1. T h e two date (t — 0, t = 1) economy contains a riskless asset ("cash") i n zero net supply and a risky asset ("stock") i n a net supply of 1 share. 2. Investors have logarithmic u t i l i t y functions and are initially endowed only 3  w i t h stock. 3. T h e two aggregate investors agree that the t = 1 value of the stock w i l l be either A or B but they differ i n the probabilities they assign to each payoff. 4. Investors act as atomistic price takers maximizing the expected u t i l i t y of terminal wealth. 5. There are no transaction costs or taxes, and asset shares are infinitely d i visible. 6. T h e riskless rate is exogenous, and there is unrestricted borrowing and lending at the riskless rate. For simplicity, we assume without loss of generality that the riskless rate is zero. The model was also formulated for the more general power utility ^(w) , where J—y| > 1. The results qualitatively the same. 3  7  94  7.  Investors do not attempt to infer each other's beliefs from prices.  4  A s shown i n Figure I I I . l , the price of the stock is P i n a short sale constrained economy, and P  u  i n an unconstrained economy. T h e two investors b o t h believe  that the stock's terminal value w i l l be either A ox B.  T h e only difference is  that the optimistic investor assigns a higher probability to the state i n w h i c h the stock's price goes up than does the pessimistic investor (i.e., TT > n ). 0  The  p  optimistic and pessimistic aggregate investors' endowments are made up of stock (X , 0  X ). p  . A Short Sale Constrained Economy W e assume that the short sale constraint is binding for the pessimistic investor. T h e optimistic investor maximizes expected t — 1 wealth. portfolio.  A t t = 0 he decides on the amounts of cash and stock i n his  Since we have assumed that the riskless rate is zero, cash does not  earn interest. T h e optimistic investor's m a x i m i z a t i o n problem is  Max  ln(W)  X >0  =  Max  7r \n[PX 0  + X (A-P)]  0  (III.l)  0  o  +(1  -  TT ) 0  In [PX  + X (B  0  0  - P)]  and similarly for the pessimistic investor's problem. T h e first order condition for the optimistic investor is  *.(A-P) PX  0  (1_-Q(B-P)  + (A - P)X  0  PX  + (B-  0  (  m  2  )  P)X  a  T h e pessimistic investor would like to hold a negative amount of the stock but cannot do so because of the short sale constraint. condition of the pessimistic investor is negative at X  = 0.  p  *M-P)  +  ( i - « M B - p  PXp  T h a t is, the first order  )  <  0  (  n  l  3  )  PXp  T h e condition for a binding short sale constraint can be written as,  (nw)  », < £ f T h e optimistic investor holds the t o t a l supply of stock, meaning X  D  = 1. In  equilibrium, the supply of stock equals the demand, so the equilibrium price is derived from the first order condition of the optimistic investor (eq. III. 2). 4 I.e.,  investors give infinite weight to their prior probabilities.  95  E q u i l i b r i u m requires the stock price to be i n the range B < P < A. In this range the unique equilibrium price is given by P = T-  VT -F  7T X )  + B(l~  (III.5)  2  where,  T  A(l  =  -  0  0  2(1 F  Xp(l  - X  0  -  7T )) 0  (III.6)  )  =  (III.7)  (1 -  Xo)  w i t h the condition that, *o >  (IIL8)  P r o p o s i t i o n 1 The price of the stock in a constrained economy P is a concave function of the two possible payoffs, A and B. P r o o f . It should be clear that T and F are increasing linear functions of A and B and that T  2  > F. We may express T and F as follows:  T = a+ 1  hx;  F = a + bx 2  (III.9)  2  where x represents A or B. T h i s means that that,  dP _ dT  2Tg -  dx  2 v !" - F  dx  7  g  (III. 10)  2  Since the price is increasing i n A and B then we must have dT 2T<f -f— > —f x  dx  5  , III.ll  d x  2^/T - F  .  2  Note that  = & dx dx  >  -f  d  dx  (I.U2)  For the more general case of u(w) = ^w there are pathological cases where for high risk aversion 7, an increase in A actually reduces the price of the underlying asset (see Fishburn and Porter, 1976). Throughout the analysis we ignore these cases. 5  1  96  T h i s is because T  > F for a l l x. while b o t h T and F are increasing linear  2  functions of x. Thus, T (x)  is a parabola, and F(x) is a straight line, and the  2  difference between T  and F increases w i t h x. Therefore, since b o t h sides of eq.  2  ( I I I . l l ) are positive, by making squares of b o t h sides, the inequality is preserved. T h i s condition is represented by  dT\  (2T———)  2  2  s) >  > 0  (IIL13)  T h e second derivative of P w i t h regard to x is given by:  4. ^  0 (*L\ I dx) 2  dP  dT  (\  dx  dx  \2j  2  2  2  2  Z  L  -  dx  L  2  dx  2  V  JTi  -  1  T  r  F -  dF\ (?T^ dT - 4£\ K* dx dx ) 1  ^'d7—57 'fe~fe 22^v/ T ' T^-FF  1 1  J  2  2  - F  2  (III. 14) Considering that T and F are linear functions of x, resulting i n second derivatives vanishing, this expression yields  J2  ^  dx  P  =  _ (dry + U  J  \ ^-^) 21  X^ " 2  F  )  ~  (in.15)  VT - F  2  2  B y eq. (III. 13) this is obviously negative.  •  Relaxing the Short Sale Constraint In an economy where short sales are allowed, b o t h aggregate investors would be able to choose their most preferred portfolios. T h e first order conditions for the stock holdings chosen by the two aggregate investors are w.r.t. X a  PX U  0  w.r.t.  + (A-  P )X u  PX  + {B-  P )X  PX  + (B-  P )X  U  a  0  U  Q  X  p  * (A-n U  + (A - P )X U  P  _„  +  R  PX  U  P  P  (  n  U  7  )  U  P  T h e market clearing condition is X + Xp = 1 0  w i t h the condition that, 97  (111.18)  (111.19) Solving for the equilibrium price, we have pnc  _  AB A(\-  -  X 7T 0  0  + B(X TT  Xp-Kp)  Note that for the case of 7r =  0  +  0  XpTTp)  the pessimistic investor does not short  p  sale, but simply sells a l l of the endowment X  i n the market. I n this case the  v  constrained and unconstrained economy yield the same price b y definition (simply b o t h first order condition are satisfied, and X = 0). p  Proposition 2 The price in a unconstrained economy is lower than the price in a constrained economy, i.e., P > P . u  Proof. T o show that the price i n the unconstrained economy is lower, note that dP d*P  u  AB(A - B)X + B(X 7r P  (A(l-X n -X 7rp) 0  0  p  0  + Xpnp)  2  0  Thus, the price monotonically increases w i t h TV . T h a t means that for ir < P^-B u  P  -B  p  the price i n the unconstrained economy is lower.  •  Proposition 3 The price in an unconstrained economy P is a concave function u  of the two possible payoffs, A and B.  Proof. W e first not that 0 < X n 0  form the fact that X + X Q  p  + X 7r  0  p  <  p  1. T h i s result is derived easily  = 1, while 7r < 1 a n d TT < 1. Thus, we can write P  0  the expression for the price as, AR  P =  — Ah + Bk  v  (111.20) ;  2  where 1 > ki,k  > 0  2  T a k i n g first a n d second derivatives: g n dP ' dB U  k,A (Afci+Bfca) 2  ^ n dP ^ ' dB 2  2  U  -2A fcnfc, (Ak!+Bk )  = T^Sfe*  2  3  2  n  ^  98  U  -  > °> I f = J0Z$fc  <  3.3  V o l a t i l i t y Effects In this section we analyze the effect of the removal of the short sale constraint  on the price v o l a t i l i t y of the underlying asset. W e analyze multiple repetition 6  of the one-period model, which allows us to derive the i n t u i t i o n on volatility changes purely from the analysis of the price of the stock as a function of the A and B values. Because our objective is deriving the i n t u i t i o n i n a simplified approach, we do not construct a general dynamic model, but rather assume that investors have one-period horizons and a new generation of investors enters the market each period. T h i s is reflected i n the following assumptions. 1.  E a c h period new values of A and B are learned. These values reflect the arrival of news. T h e difference A — B can be considered a measure of the degree of uncertainty that prevails i n the economy that period.  2.  Investors' end of period wealth is given i n terms of shares and/or cash. E n d of period wealth is consumed immediately and there is no inter-temporal substitution.  3.  T h e beliefs of each new generation of investors, as reflected i n the probabilities that they assign to the A and B values i n each period, are constant and represent the degree of divergence of beliefs i n the economy.  Proposition 4  The price as a function  concave in the constrained  7  of the future A and B values is more  economy than in the unconstrained  economy.  P r o o f . Lets assume B is constant and analyze the price as a function of the uncertain A value. F r o m Proposition 2 we know that the price is lower i n the unconstrained economy. W e also know that the two price functions (constrained and unconstrained) must intersect at two extremes values. F i r s t , i f B = A then there is no uncertainty i n the economy and the price is P = A — B.  Second,  when A is high enough, even the pessimistic investor would not want to short sale, i.e., ir > ^ | p  so a constrained and unconstrained economy yield the same  price. Since by P r o p o s i t i o n 1 and Proposition 3 b o t h price functions are concave, it must be that the constrained economy is more concave. 6  •  If we normalize the prices the analysis also holds for the return volatility.  Our results are qualitatively similar for any learning process for the beliefs of the investors. The important factor that drives the result is that there is some degree of heterogeneity in the beliefs between the two types of investors. Our assumption of single-period investors, which implies no learning, is for convenience. However, the implication that disagreement does not disappear over time seems quite realistic to us. 7  99  A s a first step i n understanding how arrival of news affects volatility, we assume that each period B remains constant, while A follows a b i n o m i a l distribution given by  prob = A :  A= A+ A  prob = 1 - A :  A = ~A- A  (111.21)  T h e binomial assumption is for simplicity but without loss of generality w i t h respect to the qualitative results. T h i s distribution simplifies the analysis of the effects of variation i n A. Figure III. 2 illustrates how varying A affects the stock price, where P is the price of stock i n the constrained economy and P  u  is the  stock price i n the market w i t h no short sale constraints. In b o t h economies the stock price is an increasing concave function of A (and of B) and the crucial difference, as stated i n Propostion 4, is that the concavity is greater i n the constrained economy t h a n the unconstrained economy. For further simplification we have drawn Figure III. 2 so that the price function i n unconstrained economy is linear. T h i s assumption leads to easier understanding of the phenomenon we describe below, but the situation is qualitatively similar i n the general case. T h e important point is that the concavity of the stock price as a function of A i n the unconstrained economy is less t h a n the concavity i n the constrained economy. To the right of the intersection between the two economies' price curves at  A , max  the short sale constraint is not binding, and eq. (III.4) does not h o l d . Denote by 8  A the value of A for which the two price functions have the same slope. Figure c  III.2 illustrates two possible cases of binomial distributions for A. T h e first case concerns A-values to the left of A , and the second case concerns A-values to the c  right of A . T h e tangent passing through point C is parallel to the price curve of c  the unconstrained economy. I n the first case, A takes two possible values: Ad + A and Ad — A. It is apparent from Figure III.2 and the above equations that i n this case the constrained economy exhibits greater stock price volatility. T h e slope of the price function is greater i n the constrained economy at the two possible values of A, resulting i n a larger price difference. Therefore, relaxation of the short sales constraint leads to a reduction i n volatility i n this case. E v e n without the binomial distribution assumption, we observe that the volatility unambiguously decreases when a l l possible values of A are smaller then A . T o the right of point c  A  c  the slope of the price function of the constrained economy is smaller t h a n  that of the unconstrained economy. Therefore, when A varies around A , but is u  always above A , then the unconstrained economy is more volatile. These results c  For values of A above A , max  holding B fixed, even the pessimistic investor is not prepared to sell short. 100  indicate that when B is constant the behavior of the volatility depends o n the range over which A varies. Depending on the distribution of A, the relaxation of short sales constraints can either increase or decrease the stock price volatility. If A takes on relatively high values i n the unconstrained economy (and B is constant) then the unconstrained economy is more volatile. O n the other hand, i f the A-values i n the unconstrained economy are relatively low (and B is constant) then the unconstrained economy is less volatile. T h e case of a varying B w i t h a constant A is illustrated i n Figure III. 3 and is similar to the previous case. T h e price function curves i n the two economies behave i n the same manner as w i t h variation i n A.  B o t h price curves are increasing functions of B. However, the  price curve i n the constrained economy is more concave. For lower values of B (and a constant A) the volatility of the stock price i n the unconstrained economy is smaller, whereas for higher values of B (and a constant A) the volatility of the unconstrained economy is greater.  W h e n we allow A to vary while keeping B constant, we also vary the magnitude of the difference A — B. T h i s difference determines whether relaxation of the short sale constraint increases or decreases the volatility of the stock price. V a r i a t i o n i n A w i t h a constant B increases volatility when A — B is high, while the opposite is true for low A — B values. O n the other hand, a variation i n B and a constant A reduces volatility when A — B is high, while the opposite is true for low A — B values. Thus, when allowing b o t h A and B to vary, we have to look at two aspects of the variability of the news. F i r s t , which variation, that i n A or that i n B, dominates? Is the news concerning the good state of the economy more uncertain or the news concerning the b a d state of the economy more uncertain? Second, what is the difference between the low A and the high B values, i.e., the m i n i m u m A — B difference? If we know this information, we can determine whether relaxation of the short sale constraint w i l l increase or decrease volatility. T h i s is summarized i n Table 1. 101  Table 1 small (Ai — B )  large (A\ — B )  AB  AB  2  small  2  small  large  N o Effect  Increase  AA  small  large  small  N o Effect  Decrease  large  Increase  N o Effect  AA large  Decrease  N o Effect  I n Table 1 we let A take two possible values: A i a n d A  (where A  2  while B takes two possible values: B i a n d B  2  (where B  2  2  > B\).  > Ai),  W i t h these  four variables we can define all of the necessary information. If A A =(A  2  Ai) > (B  2  —  — Bi) = AB the variation i n A dominates the variation i n B, while  if AA < AB the variation i n B dominates the variation i n A. A measure of the overall difference between the good and bad states is Ai — B . 2  the variables AA,  AB,  and Ai — B  2  W e allow each of  to be either small or large to derive eight  possible cases. Figure III.4 illustrates the two cases of increased volatility. In the first case, AB  > AA so the uncertainty about the b a d state dominates. Since (Ai — B ) 2  is small, we are i n the range to the right of the tangent point B  c  i n Figure III.3  and therefore relaxation of the short sale constraint increases volatility. In the second case, A A > AB so the uncertainty about the good state dominates. Since (Ai—B ) 2  is large, we are i n the range to the right of the tangent point A i n Figure c  III. 2 and therefore relaxation of the short sale constraint again increases volatility. If A A = AB  t h a n the relaxation of the short sale constraint has no effect on  volatility because the A and B variations offset each other no matter what the difference A i — B  2  is. T h e remaining two cases of decreased volatility can be  analyzed similarly. Observe that since we derive the volatility implications from the two price curves of the risky asset, different assumptions about the specific beliefs or endowments processes would not result i n qualitatively different results. A s long as there is some investors i n the economy whose beliefs are such that they want to short sell the stock, there i s , a difference i n the concavity between the constrained and unconstrained economies, which implies a change i n volatility. T h e preceding analysis implies that the economic conditions at the time of the relaxation of the short sale constraint is a crucial factor when considering their effect on volatility. If one knows the uncertainty that prevails i n the economy as  102  reflected by the difference A — B, the variability of A, and the variability of B, then one can make predictions on how the stock price volatility w i l l change.  3.4  E m p i r i c a l Illustration T h e previous sections showed that return volatility increases or decreases de-  pending on exogenous parameters and the variability of information (A- and B-values).  W e now describe an empirical case study analyzing how introduction  of options affected stock return volatility i n a specific market w i t h a short sale constraint.  9  Previous research has suggested that option i n i t i a t i o n is likely to reduce the volatility of the underlying asset. Options may have a stabilizing effect on the underlying asset volatility (Gjerde and Saettem 1995). Interaction between the spot and the derivative markets could result i n more efficient risk allocation, which would increase the demand for the underlying security and reduce its volatility. Options may cause a diversion of trading away from the underlying asset market to the option market.  However, an increase i n the size of the  opportunity set makes risky investments more attractive, resulting i n an increase i n liquidity of the underlying market.  Empirically, option listing is associated  w i t h an increase i n trading volume (e.g., Skinner 1989). b i d - ask spreads, which reduces volatility.  T h i s leads to lower  O n the other hand, options may  increase the quantity and improve the quality of information available about the underlying asset. Thus, the speed w i t h which new information is incorporated into security prices may increase, which could result i n an increase i n volatility. O u r theoretical model shows that i n a market w i t h a binding short sale constraint, option i n i t i a t i o n can result i n either increasing or decreasing volatility. T h e important factors governing the effect are the probability beliefs, the wealth endowment, and the variability of information (A- and  B-values).  T h e specific case we study concerns an Israeli stock market index.  Index  options were introduced i n the Israeli market i n August 1993. T h i s market is a good representative for testing the volatility effect because short selling was prohibited i n the Israeli market at the time of option introduction. W e test the hypothesis that the i n i t i a t i o n of index options is associated w i t h an increase i n Note that since we deal in this section with a particular case of option introduction, we cannot test whether option initiation in a short sale constrained economy reduced the price as predicted by our model. Conrad (1989) examines the effect of option introduction using an event study methodology around the date of option introduction and announcement and finds a permanent price increase in the underlying security. However, it is not clear whether her data set includes securities that were subject to short sales constraints at the time of option introduction. 9  103  the volatility of the index. Testing our model w i t h this hypothesis is appropriate because as summarized above, other theoretical and empirical studies of effects on volatility due to option initiation have found a reduction i n volatility. In January 1, 1992 a new stock index was introduced i n the T e l - A v i v Stock Market. T h e new index ( M A O F Index) is a capped i n d e x  10  weighted by market  capitalization of the 25 largest firms of the Israeli economy. T h e index represents approximately one t h i r d of the total market c a p i t a l i z a t i o n . Trading i n options 11  started 19 months after the introduction of the index (in August 1, 1993). D a i l y prices on the M A O F Index and the broad T e l - A v i v Stock Exchange Index were obtained from the Israel Securities A u t h o r i t y (ISA). T h e data are daily closing prices for the period 1/1/92 - 31/7/96. Index return volatility is calculated over periods starting 19 months prior to index option initiation (i.e., at the date the M A O F Index was launched) and ending at various dates after the initiation of the index options (1 year, 1.5 years, 2 years, and 3 years). T h e longer the interval, the more observations used i n the estimation procedure and the sharper are the results.  O n the other hand, the  longer the time frame, the higher the probability that other factors may influence the results. T h e short sale restriction is effectively relaxed when the options are initiated but additional effects may arise over time. For example, liquidity may increase w i t h time, as investors become familiar w i t h the new instrument.  The  result is that the power of the test is likely to be reduced w i t h a longer time frame. To determine whether the introduction of index options affected index return volatility, it is necessary to correct for other Israeli market forces, w h i c h influence the M A O F Index. T h e T e l - A v i v Stock Market Index reflects other economic developments that influence the Israeli Market. After removing the broad T e l A v i v Stock M a r k e t Index return, various specifications for conditional errors were tested on the residuals. F i r s t , the entire sample was checked for day of the week and month of the year effects . Second, each time frame was checked for A R M A 12  model specification up to [p=2, q=2]. I n an attempt to maintain a balance between models, a common criterion was used: the Schwarz criterion. T h e Schwarz criterion was used because it delivers relatively parsimonious specifications and is widely used i n the literature . After jointly removing the T e l - A v i v Stock M a r k e t 13  10  1 1  W i t h an upper limit of 9.5% for any single stock in the index. The index market capitalization value as of June 30, 1998 was 28,100 Million US $.  It was found that February has some significant positive effect on the return. However, this signifi disappeared after the removal of A R M A specifications. 12  13  I n all time frames the ARMA(1,1) specification was chosen. 104  and A R M A effects, an A R C H test was conducted. In a l l cases the test suggested that there was hetroscedasticity i n the errors. A joint estimation procedure was conducted including the T e l - A v i v Stock M a r k e t Index return, the A R M A ( 1 , 1 ) , and different variance models: G A R C H and G J R (up to G J R ( 2 , 2 ) ) . E s t i m a t i o n was done by maximizing the likelihood function w i t h the Schwarz criterion employed for selection. In a l l cases, the simple G A R C H ( 1 , 1 ) model was chosen and the asymmetric G J R component was not significant. Finally, i n order to check for the option i n i t i a t i o n effect, a d u m m y variable was introduced i n the variance equation for the time following the option i n i t i a t i o n date.  105  Thus, the tested specification was  aiTR  t  + a R -i 2  + o ^ t - i + e*  t  0 + ^Dop + 0e_! + / V - i 2  O  Rt-MAOF  2  2  Index return,  c r - V o l a t i l i t y of the M A O F Index R e t u r n , 2  Table 2 reports the estimation results for the coefficients, the p— values for the coefficient 0  X  of D , op  and the results of different tests on the residuals. T h e table  also includes the standard deviation and p— values when using robust standard errors . 14  T h e values of 0  X  indicate that option i n i t i a t i o n had a positive influence on the  volatility for time frames less t h a n three years . W i t h a time frame of 3 years, 15  0!becomes insignificant. In tests using simple A R C H models the significance of the option d u m m y variable was extremely high, suggesting that the G A R C H (<T _ ) component captures much of the effect of option initiation. O u r conclusion 2  x  is that option i n i t i a t i o n increased index return volatility, at least i n the short run.  A rough approximation is that the i n i t i a t i o n of option trading accounted  for about 7.5% of the unforecastable v o l a t i l i t y . 16  The standard errors are robust in the sense that conditional normality of the errors is not assumed. See BoUerslev (1986). 14  15  Results for greater than three years were also insignificant.  Looking at the 1.5 years time frame, the GARCH model coefficients should add up to 1 if they capture all the volatility. Adding them up gives 0.908. The other 10% of volatility is explained by the average (/3 ) and the option initiation variable (Pi). Since (3 ~ 3/3 it means that the option initiation component is responsible for approximately 7.5% of the unforcastable volatility. 16  0  1  106  0  Table 2 GARCH(1,1)  Rt = a{TR  t  + a P _ ! + a e _i + e 2  t  3  o\ = Po + PiDap + P e t-i 2  2  t  t  + /V  2  -i  1 Year  1.5 Years  2 Years  3 Years  Po  0.479£ - 6  0.103.E - 5  0.724£ - 6  (0.043.E-5)  (0.267£-6)  0.19LE-6  Pi  0.172£ - 5  0.316£ - 5  0.115£-5  (0.25OE-6)  (0.096.E-5)  (0.139B-5)  (0.053.E-5)  (0.091.E-6)  0.886£ - 7 (0.850E-7)  p—value for Q  0.074  0.022  0.029  0.298  Pi  0.085  0.124  (0.025)  (0.032)  0.137  0.073  Ps  0.870  0.784  0.808  0.916  Residuals Tests  Y E S : C a n reject hypothesis that residuals are normal.  B o x - Pierce  No  No  No  No  Goodness of F i t  Yes  No  No  Yes  Jarque - B e r a  Yes  Yes  Yes  Yes  0.12LE- 5  0.175£-5  0.059£ - 5  0.119£-6  0.156  0.071  0.0515  0.458  x  (0.040)  (0.059)  (0.028) (0.039)  (0.014) (0.016)  Robust Standard Errors P STD 1  p-value 3.5  for Q  x  S u m m a r y and Conclusions T h i s study examines the influence of short sale restrictions o n the pricing  behavior.  Considering a market subject to a short sale restriction, so long as  investors disagree o n the probabilities assigned to final payoffs and so long as some investors would like to short sell the risky asset, the equilibrium price of the risky asset is higher and more concave t h a n i n an unconstrained economy. W e focus o n the effect of relaxation of the short sale constraint have o n the 107  price volatility of the risky asset. W e show that generally there is a range of i n creasing volatility and a range of decreasing volatility, depending on the variability of information and the other values of the exogenous variables of the economy. These results contradict the belief that relaxing the short sale constraint should always increase volatility due to the increasing effect of bad news. A s an empirical case study, we looked at the behavior of the T e l - A v i v stock market, i n which short sales are prohibited, when index options were introduced i n August, 1993. T h e data indicate that the return volatility of the index i n creased after options were introduced.  108  REFERENCES CITED  1. A s q u i t h , P. and Meulbroek, K . , 1995. A n E m p i r i c a l Investigation of Short Interest. W o r k i n g Paper. Boston, Mass., H a r v a r d Business School. 2.  Black, F . , and Scholes, M . , 1973. T h e P r i c i n g of Options and Corporate Liabilities. Journal of P o l i t i c a l Economy 81, 637-659.  3.  Bollerselv, T . , 1986. Generalized Autoregressive C o n d i t i o n a l Heteroskedasticity. Journal of Econometrics 31, 307-327.  4. Bollen, N . , 1998. A Note on the Impact of Options on Stock R e t u r n Volatility. Journal of B a n k i n g and Finance 22, 1181-91. 5.  Cao, H . , 1999. T h e Effect of Derivative Assets on Information A c q u i s i t i o n and Price Behavior i n a R a t i o n a l Expectations E q u i l i b r i u m . Review of F i n a n c i a l Studies 12, 131-163.  6.  C h a t r a t h A . , Ramchander, S., and Song, F . , 1995. Does Options T r a d i n g L e a d to Greater C a s h M a r k e t Volatility? Journal of Futures Markets 15, 785-803.  7.  Chaudhury, M . , and Elfakhani, S., 1997. L i s t i n g of P u t Options: Is there any V o l a t i l i t y Effect? Review of F i n a n c i a l Economics 6, 57-75.  8.  C o n r a d , J . , 1989. T h e P r i c e Effect of O p t i o n Introduction. J o u r n a l of Finance 44, 487-498  9. Damodaran, A . , and L i m , J . , 1991. T h e Effect of O p t i o n L i s t i n g o n the Underlying Stocks' R e t u r n Processes. Journal of B a n k i n g and Finance 15, 647664. 10.  Danielsen, B . , and Sorescu, S., 2001. W h y D o O p t i o n Introduction Depress Stock Prices?  A n E m p i r i c a l Study of Diminishing Short-Sale Constraints.  Journal of financial and Quantitative Analysis 36, 451-484. 11.  Detemple, J . , and Jorion, P., 1990. O p t i o n Listing and Stock Returns: A n  E m p i r i c a l Analysis. Journal of B a n k i n g and Finance 14, 781-801. 12.  D i a m o n d , D . , and Verrecchia, R . , 1987. Constraints on Shortselling and A s -  set P r i c e Adjustment to Private Information. Journal of F i n a n c i a l Economics 18, 277-311. 13.  F i s h b u r n , P., and Porter, R . , 1976. O p t i m a l Portfolios w i t h One Safe and  One R i s k y Asset: Effects of Changes i n Rate of R e t u r n and Risk. Management Science 22, 145-162. 109  14.  Gjerde O . , and Saettem, F . , 1995. O p t i o n Initiation and U n d e r l y i n g M a r k e t  Behavior: Evidence from Norway. Journal of Futures Markets 15, 881-889. 15.  Grossman, S., 1988. A n analysis of the Implications for Stock and Futures  Price Volatility of P r o g r a m Trading and D y n a m i c Hedging Strategies. Journal of Business 62, 211-235. 16. Hakansson, N . , 1978. Welfare Aspects of Options and Supershares. Journal of Finance 33, 759-776. 17. Huang, J . , and Wang, J . , 1997. Market Structure, Security Prices and Informational Efficiency. Macroeconomics-Dynamics 1, 169-205. 18. Hollifield, B . , and Gallmeyer, M . , 2002. A n E x a m i n a t i o n of Heterogeneous Beliefs w i t h a Short Sale Constraint, W o r k i n g Paper, Carnegie M e l l o n University. 19. Jarrow, R . , 1980. Heterogeneous Expectations, Restrictions on Short Sales, and E q u i l i b r i u m Asset Prices. Journal of Finance 35, 1105-13. 20. K r a u s , A . , and Smith, M . , 1996. Heterogeneous Beliefs and the Effect of Replicatable Options on Asset Prices. Review of F i n a n c i a l Studies 9, 723-756. 21.  L o n g , M . , Schinishki, M . , and Officer, D . , 1994. T h e Impact of O p t i o n L i s t i n g  on the Price Volatility and Trading Volume of Underlying O T C Stocks. J o u r n a l of Economics and Finance 18, 89-100. 22.  M i l l e r , E . , 1977. R i s k Uncertainty, and Divergence of O p i n i o n . Journal of  Finance 32, 1151-68. 23.  Ross, S., 1976. Options and Efficiency. Quarterly Journal of Economics 90,  75-89. 24.  Skinner, D . , 1989. Options Markets and Stock R e t u r n Volatility. J o u r n a l of  F i n a n c i a l Economics 23, 61-78. 25.  St. Pierre, E . , 1998. T h e Impact of O p t i o n Introduction on the C o n d i t i o n a l  R e t u r n D i s t r i b u t i o n of U n d e r l y i n g Securities. F i n a n c i a l Review 33, 105-118. 110  P e s s i m i s t i c Investor  Optimistic Investor  t = 0  p  t = 1  t = 0  P  (P ) u  1-n.  t = 1  (P  u  B  Figure I I I . l : Scehmatic description of the two aggregate investors' beliefs about future stock payoff (TTO > TT ). T h e price i n a constrained economy is P , while the price i n an unconstrained P  economy is  P. u  Ill  Figure III.2: Schematic description of the price as a function of the A-values i n the constrained and unconstrained markets. T h e P curve is for the constrained economy, where as the P  u  curve is for the unconstrained market, where options are traded.  112  Figure III.3: Schematic description of the price as a function of the B-values  i n the con-  strained and unconstrained markets. T h e P curve is for the constrained economy, where as the P  u  curve is for the unconstrained market, where options are traded.  113  Panel A  Panel B  Aj  A2  A  c  Figure III.4: T w o cases of increasing volatility. Panel A illustrates a case where AA > and the difference A\ — B  2  difference A\ — B  2  is large. Panel B illustrates a case where AA  is small.  114  < AB  AB  and the  

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