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Three essays on investments and incentives : an application to pension plans Soumare, Issouf 2005

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T H R E E E S S A Y S O N I N V E S T M E N T S A N D I N C E N T I V E S : A N A P P L I C A T I O N T O P E N S I O N P L A N S by I S S O U F S O U M A R E B . S c , Universite Nationale de Cote d'lvoire, 1992 M . S c , Universite Nationale de Cote d'lvoire, 1993 M . A . , E N S E A - A b i d j a n , Cote d'lvoire, 1996 M . S c , Universite Laval , Canada, 2000 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S Business • we accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A . October 2004 © Issouf Soumare, 2004 Abstract Despite significant diversification costs, a considerable number of defined contribution pension plans have large holdings of the shares of the company they work for. The first essay presents an incentive model in which the voluntary optimal company shareholdings of the worker is derived. The argument is that the worker voluntarily hold shares of his employer to benefit from his ability to adjust effort. In addition, the analysis provides a separate role for senior managers relative to other employees. The manager takes actions that influence the productivity of the entire work force. The worker on the other hand influences output by exerting effort. The optimal effort level w i l l take into account the productivity decision of management while the productivity decision takes into account the worker's effort. In the second essay, I develop a model to study the equilibrium implications when some investors in the economy overweight a subset of stocks within their portfolio. I find that the excess returns for the overweighted stocks are lower, a l l else being equal. The riskfree rate increases and the market price of risk of the overweighted stock de-creases, which create extra incentive for unconstrained agents to exit the stock market and hold bonds, hence clearing the market. The changes of stocks' volatilities are am-biguous. Final ly , I provide an accurate quantification for agents' welfare. I also discuss the implications of my model in the context of defined contribution pension plans. In the th ird essay, a dynamic general equil ibrium model for a two-country, two-good exchange economy in incomplete markets is developed. The model implies that when a domestic country caps foreign investment in some key industries in the domestic economy, the cost of capital of the protected industry increases, al l else being equal, that of the non-protected industry decreases. O n the other hand, when imposing restrictions on its residents' foreign investments, the domestic country improves its cost of capital , al l else being equal. Furthermore, in both restricted economies, the cost of risk free borrowing and lending is lowered. However, when domestic residents are capped in their foreign investment, the uncovered interest rate parity relationship is violated. B y artificially restricting agents' investment, countries can reduce financial contagion effects because stock markets are affected asymmetrically. Th is result contributes to the debate on why recent crises in international financial markets have had different effects on countries located in same geographical area or having similar economic characteristics. The effects of the restrictions on stock market volatilities are ambiguous. Final ly , we show that when the restriction is protective, the welfare of the agents of the country imposing the restriction increases. This result helps us understand why some countries are so reluctant to change their protective financial policies. Table of Contents Abstract ii Table of Contents iv List of Tables vi List of Figures vii Acknowledgments viii 1 Introduction 1 2 Essay 1: Incentives and Voluntary Investment in Employer Shares 7 2.1 Abstract 7 2.2 Introduction 7 2.3 Empir i ca l evidence on D C plans investments 10 2.4 Incentive - voluntary investment model 12 2.4.1 Information and price dynamics 13 2.4.2 Workers' portfolio choice and effort level 13 2.4.3 The manager's compensation and firm productivity 18 2.5 Conclusion 20 2.6 Appendix 22 3 Essay 2: Excessive Holdings and Equi l ibrium Asset Prices 24 3.1 Abstract 24 3.2 Introduction 24 3.3 The economic setting 28 3.3.1 Securities market and information structure 28 3.3.2 Agents 29 3.4 Equi l ibr ium in a pure-exchange economy 33 3.4.1 The benchmark economy - no excess holdings case 34 3.4.2 The restricted economy - excessive holdings case 36 3.5 Restricted economy wi th logarithmic preferences 41 3.5.1 Returns and risk premia 41 3.5.2 Stock prices and returns' volatilities 45 3.5.3 Consumptions and welfare 46 3.6 Conclusion 48 3.7 Appendix 50 4 Essay 3: International Diversification, Investment Restrictions and T h e Exchange Rate 58 4.1 Abstract 58 4.2 Introduction 58 4.3 The model 63 4.4 The Bench Market Case - no restriction on assets holding 65 4.5 Equ i l i b r ium wi th Investment Restrictions 69 4.5.1 Investors' optimization problem 69 4.5.2 Restriction on foreign investors 70 4.5.3 Restriction on domestic residents 81 4.6 Conclusion 90 4.7 Appendix 92 Bibliography 116 List of Tables 1.1 Aspects of U . S . Private Sector Pension Plans: 1985-2001 2 1.2 Percentage allocation of retirement plans assets in company stock as of year 2001 3 2.1 Percentage allocation of retirement plans assets in company stock as of year 2001 10 4.1 Simulated correlations between domestic and foreign equity re-turns 79 4.2 Simulated correlations between domestic and foreign equity re-turns 89 List of Figures 3.1 Equi l ibr ium riskfree rate 56 3.2 Market equity premium 56 3.3 Consumption paths 57 3.4 Welfare variation 57 Acknowledgments I would like to thank Tan Wang (my supervisor) and Ron Giammarino for their constant support and help during this project. I am thankful to my other committee members, Murray Frank, Rob Heinkel and A l a n Kraus for their guidance and support, my external examiner Domenico Cuoco, my two university examiners Patrick Francois, R a l p h Winter for their insightful comments and suggestions. Also thanks to the faculty and P h . D . students at the Sauder School of Business at U B C , V a n Son L a i , and al l the people I have met during my P h . D . years. P a m also grateful to the Faculty of Graduate Studies and the Sauder School of Business of The University of B r i t i s h Columbia for their financial supports. Of course, I am grateful to my parents and my wife for their patience and love. Wi thout them this work would never have happened. v in To M y M o m K a n d i a Camara 1 Introduction This thesis consists of three essays that focus on investments in pension plans and inter-national capital markets. Specifically, in the first two essays I examine questions related to the investments made to support pension plans and the equilibrium implications of those decisions. In the th ird essay I study the equilibrium properties of international capital markets when investors face investment restrictions. I now give the motivation for the first two essays, and later I provide the motivation for the th i rd essay. I also explain how the three essays are related. In essence, pension plans are vehicles that provide post employment income for work-ers. They typically take two broad forms, defined benefit (DB) and defined contribution ( D C ) . 1 The D B plan is the tradit ional plan and consists of a promise to pay workers a predefined amount, based on their years of service and their wage history. Employ-ers contribute to an investment pool that is used to fulfill the promised benefits. A n y shortfall in investment income relative to promised payments is the responsibility of the employer. Hence, risk is largely born by the employer and accordingly the employer has control of the investment decision. Al though interesting, the management of D B plans is not the subject of my thesis. I focus on D C plans. Here, the worker and her employer contribute to a fund that belongs to the worker. The retirement income is the outcome of returns to this specific investment portfolio. These funds were originally created as means of profit sharing. The employer matches any contributions from workers according to a predetermined formula. The contributions from the employer can be in cash or shares. The post employment income fluctuates wi th the investment returns of the plan so that, in contrast to the D B plan, the risk rests w i th workers. Accordingly, investment decisions are mainly made by workers. These investment decisions by workers are essentially the subject of the first two essays. Over the last 20 years, D C plans have gradually replaced D B plans as shown in Table 1See Mitchell and Utkus (2002) for an excellent overview of the history of pension plans. 1.1. This shift has taken place despite the fact that most D B plans in the U .S . , are guaranteed by the Pension Benefit Guaranty Corporation ( P B G C ) , D C plans are not. Huberman and Sengmueller (2004) report that at the end of 2000, D C plans (private and public) had 2.5 tr i l l ion dollars in total assets under their control, equivalent to about 20% of the total market capitalization of the New York Stock Exchange. Table 1.1: Aspects of U . S . Private Sector Pension Plans: 1985-2001. A . Number of Pension Plans Year Total D B Plans D C Plans T3S5 632,135 170,172 461,963 1990 712,308 113,062 599,245 1995 693,404 69,492 623,912 1998 730,031 56,405 673,626 2001 e 758,000 51,000 707,000 B . Number of Act ive Pension P l a n Participants (000) Year Total D B Plans D C Plans T5BS 6%253 29,024 33,244 1990 61,831 26,344 35,488 1995 66,193 23,531 42,662 1998 73,328 22,994 50,335 2001 e 78,000 22,500 55,500 C. Pension Plans Assets ($ millions) Year Total D B Plans D C Plans T58S" 1,252,739 826,117 426,622 1990 1,674,139 961,904 712,236 1995 2,723,735 1,402,079 1,321,657 1998 4,021,849 1,936,600 2,085,250 2001 e 4,000,000 1,900,000 2,100,000 Source: Reproduced from Mitchell & Utkus (2002, Table 1, p.36) The empirical literature has widely documented many special features inherent to D C plans assets allocations. In fact, according to Benartzi (2001), a th ird of the assets in large D C plans are invested in the stock of the f irm that employs the worker, so called company stock. A quarter of workers' discretionary contributions are invested in company stock. Table 1.2 exhibits the percentage of company share in some major D C pension plans. Moreover, Mitche l l and Utkus (2002) report that workers invest more in company Table 1.2: Percentage allocation of retirement plans assets in company stock as of year 2001. Company % company stock Company % company stock Procter &;Gamble 94.7% Wi l l iams 75.0% Sherwin-Wil l iams 91.6% McDonald 's 74.3% Abbott Laboratories 90.2% Home depot 72.0% Pfizer 85.5% McKesson H B O C 72.0% B B & T 81.7% M a r s h & M c L e n n a n 72.0% Anheuser-Bush 81.6% Duke Energy 71.3% Coca-Cola 81.5% Textron 70.0% General Electric 77.4% Kroger 65.3% Texas Instruments 75.7% Target 64.0% W i l l i a m Wrigley, Jr . 75.6% Household Int ' l . 63.7% Source: Reproduced from Purcell (2002, Table 1, p.4) (from D C Plan Investing, Institute of Management and Administration, N Y ) . stock when the company has experienced a price run up, and they rate the firm's stock as less risky than identical individual stocks. Benartzi and Thaler (2001) find that workers follow the so-called '1/n naive diversification strategy', by dividing their contributions evenly across the funds offered in the plan, treating company shares as one investment category even though it is far less than 1/n of the market. A n d , when the employer's contributions are automatically directed to company stock, employees invest more of their own contributions in company stock, perhaps because they interpret the allocation of the employer's contributions as implicit investment advice. Overall , according to L iang and Weisbenner (2002) it seems that the plan design (number of investment alternatives, employer match in company stock) has a strong influence on D C plan assets allocation in company stock. M a n y people believe that the holding of company shares by workers is imposed to them by their company management, this is not the case. Despite the lack of diversifi-cation suffers by workers, through their large holding of company shares, the majority of company shares in D C plans are voluntarily held by employees. Whi le the Employee Retirement Income Security Act ( E R I S A ) of 1974 restricts the investment of D B plans in stock or real estate of the employer to 10% of total assets, D C plans are exempted from this rule. Erickson (2002) reports that "In a February 2002 pol l of 1000 individuals con-ducted by P u t n a m Investments, 57 percent of investors who held company stock in their 401 (k) accounts were opposed to legislative efforts to restrict the amount of company stock in defined contribution plans; only 20 percent favored such restrictions." It is likely that some workers would oppose restrictions on the amount of company stock in D C plans. In Huberman and Sengmueller (2004), it is quoted that "For instance Motoro la eliminated its policy l imit ing its employees' investment in Motorola stock to 25% of their contributions after employees complaints". The apparently puzzling investment behavior by plan participants has drawn a greater level of academic interest in the characteristics and structure of D C plans. Unfortunately, these papers depict workers as naive or irrational investors who follow the lead of senior managers. Cho i et al . (2001) have even qualified workers as followers of 'the path of least resistance', and Huberman and Sengmueller (2004) have suggested the existence of a corporate culture. In the first essay, I examine the optimal portfolio allocation of worker's D C pension wealth using an 'investment-incentive' model. I propose a partial equil ibrium model to examine how workers' D C pension plan assets are allocated and how this allocation is related to the productivity of the firm. Given the magnitude of D C plans investments, it is clear that the large shareholdings w i l l have implications for prices in the economy. In the second essay, I develop a general equil ibrium model to analyze the impact the large shareholdings has on stock prices and wealth. Since my objective is to study the equilibrium characteristics, I abstract from the incentive and information properties analyzed in the first essay and impose exogenously the holding constraint. Although this is a simplification from reality, the model provides new insights. The model gives new formulation for the cross-sectional test of stock returns especially for companies offering D C plans. Recent crises on international capital markets have reopened the debate on whether or not capital market liberalization is good for the countries involved. Whi le most of the international finance literature supports the idea that economic liberalization is a worthwhile goal to pursue, the international macroeconomics literature presents a mixed view. Indeed, Dumas and Uppa l (2001) and Henry (2003) laud the virtues of market liberalization. But despite the many claimed advantages, we st i l l observe governments that are reluctant to fully open their domestic capital markets even when they are well-developed. It is common to see countries artificially restrict their residents' investments abroad or l imit foreign investments in some key sectors of the domestic economy. The view that market liberalization is a worthwhile goal to pursue at any cost is not shared by eminent economists including the 2001 Nobel Prize winner in Economics, Joseph E . Stiglitz, who believes that market liberalization creates instability, increases poverty and economic insecurity in the liberalizing country. In order to provide policy makers wi th better guidance in their decision taking pro-cess, in the th ird essay I propose a general equil ibrium model of international capital markets in the presence of investment restrictions. I contribute to the following debates: (i) W h y do recent crises in international financial markets have had different effects on countries located in same geographical area or having similar economic characteristics?; (ii) Does capital market liberalization cause financial instability?; (iii) W h y are some countries so reluctant to change their protective financial policies? Even though the th ird essay is independent from the first two, the methodology used is similar to the one used in Essay two. In Essays two and three because of the exogenous constraint imposed on agents, markets are incomplete. In that context, I use a recent developed technique to solve for the equil ibrium, which gives easy economic interpretations for the results. Furthermore, the th ird essay can also be applied in pension plans investment context. Indeed, in many capital markets, pension plans face restrictions on how they should be invested. Good examples include Canadian pension fund investment regulations, which stipulate that Canadian pension funds cannot invest more than 30% of their assets under management in foreign stocks. The rest of the document is structured as follows. Chapter 2 presents the first essay "Incentive and voluntary investment in employer shares". Chapter 3 presents the second essay "Excessive holdings and equilibrium asset prices". Chapter 4 presents the th ird essay "International diversification, investment restrictions and the exchange rate". 2 Essay 1: Incentives and Voluntary Investment in Employer Shares 2.1 Abstract Despite significant diversification costs, a considerable number of defined contribution pension plans have large holdings of the shares of the company they work for. We present an incentive model in which we derive the voluntary optimal company shareholdings of the worker. The argument is that the worker voluntarily hold shares of his employer to benefit from his abil ity to adjust effort. In addition, the analysis provides a separate role for senior managers relative to other employees. The manager takes actions that influence the productivity of the entire work force. The worker on the other hand influences output by exerting effort. The optimal effort level w i l l take into account the productivity decision of management while the productivity decision takes into account the worker's effort. 2.2 Introduction Enron's financial distress resulted in both job losses and significant investment losses for employees of the company due to the huge percentage of Enron's retirement plan that was invested in Enron shares. A s of December 31, 2000, Enron shares made up 62 percent of the assets held in the company's defined contribution (DC) pension plan, Purce l l (2002). A n estimated 89 percent of these shares were purchased by employees while the balance represents the corporation's matching contribution. 1 Such a concentration of human and financial capital in one company flies in the face of the main lesson of portfolio theory. Yet , Enron employees were not alone in holding a large percentage of their pension assets in the shares of the company that employs them. In fact, a th ird of the assets in large retirement savings plans are invested in company stock, and a quarter of workers discretionary contributions are invested in company stock, •'•There are two basic types of pension plans: defined contribution (DC) and defined benefit (DB) plans. In D C plans (or 401 (k) accounts), the employer agrees to contribute a certain amount in each period to employees' pensions. The employees bear all investment risk since the firm has no obligation beyond making its periodic contributions. In D B plans, the employer promises to pay a certain amount of benefits at retirement, which depends on the employee's years of service and history of wages over the employment period. Risks are borne by the firm. Benartzi (2001). Surprisingly, these huge investments of D C pension plans assets in company stock are mostly voluntary purchases by employees themselves, and not dictated explicitly by the employer. A t the end of 2000, D C plans had 2.5 tr i l l ion dollars total assets under their control, equivalent to about 20 percent of the total market capitalization of the New York Stock Exchange, Huberman and Sengmueller (2004). Given this importance of D C pension plans some have argued for legislation to cap the amount that defined contribution pension plans are allowed to invest in company stock. Th is proposal does not, however, seem to have strong support among plan participants. Interestingly, a l -though D B plans are prevented from holding more than 10 percent of the plans assets in the stock or real estate of the employer, D C plans are exempted from this rule. Indeed, Erickson (2002) reports, "In a February 2002 poll of 1000 individuals con-ducted by P u t n a m Investments, 57 percent of investors who held company stock in their 401 (k) accounts were opposed to legislative efforts to restrict the amount of company stock in defined contribution plans; only 20 percent favored such restrictions... M a n y commentators in the financial press have noted that employees have faith in their em-ployers and do not believe that an Enron-or Adelphia , Qwest or WorldCom-catastrophe could happen to them or to the company stock that seems like, and often has been, such a good investment." Purcel l (2002) notes that, "In most majority of the company stock in 401 (k) plans represents voluntary purchases by employees. It is likely that some workers in these firms would oppose restrictions on the amount of company stock in 401 (k)." Huberman and Sengmueller (2004) state, "In general it is a bad idea to invest a high proportion of one's wealth in a single firm. It is even worse when that single f irm is one's employer... B u t individuals do it , and they like to do i t . For instance Motoro la eliminated its policy l imit ing its employees' investment in Motoro la stock to 25 percent of their contributions after employee complaints." In this paper we examine the optimal portfolio allocation of worker's D C pension wealth using an 'investment-incentive' model. Our argument is that workers hold com-pany stock to benefit from their ability to adjust effort. Indeed, since workers have the abil ity to influence the productivity of the firm, by holding shares of the company, they receive part of the profit which wi l l go to other investors otherwise. We discuss the empirical evidence suggesting that workers rely on past attributes of company stock when making their portfolio decision. They invest more in company shares if the stock has performed well over the past years. They also perceive their company stock as safer than individual stocks. Our arguments are as follow. Workers' investment in employer shares is proportional to the productivity of the company. One way for workers to assess future productivity is by measuring past f irm performance. Indeed, if the company has performed well over the last years, they w i l l infer that the manager in place is a high productivity one. Moreover, workers are firm insiders, their assessment of f irm risk is based on the conditional variance, which is lower than the unconditional variance observed by the econometrician. The basic economic forces at work in our model are otherwise similar to those found in the standard principal/agent literature pioneered by Holmstrom (1979), and Holmstrom and M i l g r o m (1987, 1991), among others. In this literature, the optimal contract imposes suboptimal diversification on workers in order to induce higher levels of effort. This has been applied directly to the case of executive compensation by Garvey and M i l b o u r n (2003), J i n (2002), and employees' compensation by Oyer and Schaefer (2002) among others. However, we depart from this literature by assuming that workers are responsible for deciding how to invest their defined contribution pension plans' assets. We argue that, when they have the ability to affect the productivity of the firm, workers w i l l invest in company stock and benefit from the rent of their own effort. In addition, our analysis provides a separate role for senior managers relative to other employees. Managers take actions that influence the productivity of the entire work force. Workers on the other hand influence output by adjusting their effort level. The optimal effort level w i l l take into account the productivity decision of management while the productivity decision takes into account workers' effort. The rest of the paper is organized as follows. Section 2.3 provides a survey of the main empirical findings on D C plans assets allocations. Section 2.4 presents our model and analyzes the voluntary stock ownership wi th respect to the firm's productivity. Section 2.5 concludes. 2.3 Empirical evidence on D C plans investments The growth in D C plans over the last decades seems to have drawn a greater level of academic interest in their characteristics and structures. Benartzi (2001) found that, a th ird of the assets in large retirement savings plans are invested in company stock, and a quarter of workers discretionary contributions are invested in company stock. Table 2.1 exhibits the percentage of company share in some major D C pension plans. Table 2.1: Percentage allocation of retirement plans assets in company stock as of year 2001. Company % company stock Company % company stock Procter&Gamble 94.7% Wi l l iams 75.0% Sherwin-Wil l iams 91.6% McDonald 's 74.3% Abbot t Laboratories 90.2% Home depot 72.0% Pfizer 85.5% McKesson H B O C 72.0% B B & T 81.7% M a r s h & M c L e n n a n 72.0% Anheuser-Bush 81.6% Duke Energy 71.3% Coca-Cola 81.5% Textron 70.0% General Electric 77.4% Kroger 65.3% Texas Instruments 75.7% Target 64.0% W i l l i a m Wrigley, J r . 75.6% Household Int ' l . 63.7% Source: Reproduced from Purcell (2002, Table 1, p.4) (from D C Plan Investing, Institute of Management and Administration, N Y ) . Benartz i also finds that when employer's contributions are automatically directed to company stock, employees invest more of their own contributions in company stock. He argues that workers interpret the allocation of the employer's contributions as implicit investment advice or an "endorsement effect". In a survey, Benartzi found that workers invest more in company stock after a price run up, and few respondents rate the firm's stock as riskier than the market. Huberman and Sengmueller (2004) found that past performance of the company is the most salient piece of information employees rely on when they make changes to their portfolio. They argue that investors in 401 (k) accounts listen to wrong signals and tend to increase their exposure to company stock if returns have been high and their employer is doing well. They do not react to negative returns, however. L iang and Weisbenner (2001) found that the plan design (number of investment alternatives, employer match in company stock, past firm performance, and firm's financial characteristics such as market-to-book ratio, price volatility, etc.) is very important in determining the share of 401 (k) assets in company stock. Mitche l l and Utkus (2002) found that company stock held in D C plans is a large firm phenomenon, and also asset allocation levels to company stock are function of plan size. Benartzi and Thaler (2001) show that some investors follow the "1/n naive diver-sification strategy": they divide their contributions evenly across the funds offered in the plan, treating company shares as one investment category even though it is far less than 1/n of the market. Based on survey research, Cho i et al . (2001) concluded that "... employees often follow Hhe path of least resistance'. For better or for worse, plan administrators can manipulate the path of least resistance to powerfully influence the savings and investment choices of their employees." Meulbroek (2002) measures the cost of non-diversification to employees when they overload their D C pension plans in company stock. She concludes that the value an employee sacrifices relative to a well-diversified equity portfolio of the same risk aver-ages 42 percent of the market value of the firm's stock under reasonable assumptions. Despite this high cost of non-diversification, employees st i l l hold their company stock even when they are not required to do so. She argues that, the encouragement by firm to hold company stock may be explicit, by prohibiting employees from selling company stock in their retirement plans, or implic it , by characterizing the firm's stock as a "good investment" or by suggesting that it provides tangible evidence of employees' loyalty. This empirical evidence largely paints employees as naive investors who are strongly influenced by senior managers of the company and typically overinvest in the company's shares. In the next section we propose an incentive-investment model that provides the optimal portfolio choice of workers as the result of optimal behavior on the part of employees and firms. 2.4 Incentive - voluntary investment model We consider a four dates model (t = 0,1, 2,3). There are three securities available in the economy: the risk-free security paying a constant return 77, the firm's stock indexed by 1, and the rest of the market portfolio indexed by m. The rest of the market portfolio (stock m) does not include the firm's stock. We assume no taxes and no transaction costs. Wi thout loss of generality, the in i t ia l supply of company stock is unity. The game is structured in the following time line. A t date t = 0, the f irm hires a manager and sets her compensation package (b,s), where b is the pay-for-performance sensitivity (PPS) and s the monetary compensation. Shareholders expect the manager to put an effort fii which w i l l increase the firm's overall productivity. A t t = 1 the company's single worker is hired in a competitive labor market at wage wo- We assume that there is a single worker in the firm. The productivity of the firm is revealed to the worker as fj,i. Having observed /x i , the worker makes his investment consequently. He allocates his pension wealth among the three securities by investing A\ in the firm's stock, Am in the market equity, and the rest in bonds. Unlike the standard Pr inc ipa l /Agent framework, here the agent (the worker), rather than the f irm (the principal) chooses the percentage of his wealth to hold in the firm. In our framework, the worker decides on his fund allocation, he decides voluntarily to hold the company stock or not. A t date t = 2, the worker learns his true type and decides on the level of his effort /, which increases the value of the firm. F ina l ly at date t = 3, the output is produced and the game ends. 2.4.1 Information and price dynamics The firm's return is a function of the productivity generated by the manager and the worker. We assume a linear functional form for the stock return wi th respect to the productivity ii\ and the effort I. Thus stock returns are ri = /ii+ §Z + ei, (2.4.1) rm = fim + em, (2.4.2) where is a risk source of stock i and is normally distributed wi th mean zero and variance of. The correlation between the two risk sources, e\ and em, is p. The random variable q captures the impact of workers' effort on company stock return, and it is normally distributed wi th mean zero and variance rf. The correlation between g a n d ej is zero. Since we are in a rational equil ibrium environment, the market price of the firm's stock at each date satisfies the rational expectation hypothesis. We use the dynamic programming approach to solve our model. Hence we first solve workers' portfolio and effort choice problem, and next we solve the incentive problem of the manager. 2.4.2 Workers' portfolio choice and effort level • Since we assume a single worker for the firm, the incentive constraint of the worker w i l l rule out the debate on free-riding. We recognize that this is a simplification from reality. Indeed, in real life situations, there should be some enforcement mechanism such as behavioral norms to avoid the free riding problem. However, even though we are in a simplified world, the model implications are useful for our understanding of workers' portfolio allocation. The worker's total wealth at t = 1, WQ, is composed of the salary and the market value of the matching contributions from the f i rm. 2 Workers invest their in i t ia l wealth 2 One implicit assumption we are making here is that when workers receive company shares as match-ing contributions from the firm, they can sell the shares on the open market or keep them without restrictions. However it is worth noticing that, some companies impose restrictions on workers from selling the matched contributions in company stock. in the available three securities: firm's stock, market equity and riskfree asset; Hence, their future wealth is Ww = A i ( l + r i ) + A » ( l + rm) + (w0 - Ax - Am)(l + rf), (2.4.3) where A\ and Am are respectively the amounts invested in the firm stock and the market portfolio. This terminal wealth expression of workers contains three terms, respectively the return amounts earned on the firm's stock, the market equity and the bond invest-ment. A t t = 2, the worker exerts the effort I. The production of the effort I incurs a cost Cw(l) to the worker. The cost function Cw(-) is an increasing convex function of I, i.e., C'w(l) > 0, C'w(l) > 0, V7 > 0. For simplicity, we use a quadratic cost function without lost of generality: CW{1) = l2/2. (2.4.4) The worker is a risk averse agent wi th constant absolute risk aversion coefficient jw • He makes his investment decision based on the distribution of q and its expectation about his effort I. Later in the paper when we solve for the effort of the worker, it can be easily shown that the return expression in equation (2.4.1) is not normally distributed. Hence the wealth doesn't follow a normal distribution. Nevertheless, we are able to solve for the worker's optimal decision by assuming that the worker makes his portfolio choice decision so as to maximize a Mean-Variance ut i l i ty of his end-of-period wealth: 3 max E[UW] = E[WW] - 0.5jwVar(Ww) - E\Cw(l)), (2.4.5) Ai,Am under his incentive constraint Aiql{q>0} = C'w(l) and participation constraint E[Uw] > Uw The effort level of the worker is obtained from its incentive constraint condition. Indeed, by putt ing an effort later, he benefits from his shareholdings of company stock. The welfare expression in (2.4.5) has two disutil ity parts: the disuti l ity for bearing risk and the disutil ity for putt ing the effort I. The disutil ity for bearing risk increases 3 W e justify this assumption of Mean-Variance utility by the fact that investors like return but dislike risk. However, since returns are not normally distributed in this framework, the maximization of the Mean-Variance utility is not equivalent to maximize a negative exponential utility of wealth. with the risk aversion coefficient of the worker. It is captured by half the variance of the portfolio times the risk aversion coefficient. The disutil ity for putt ing the effort I is captured by the cost of producing the effort. We are making the implicit assumption that the worker's risk type is diversifAable. The market and the worker know the distribution of q at date t = 1. Therefore, the prices and the worker's wage are set based on the distribution of q. Here, we take \x\ as exogenously given and it represents the firm's specific productivity. In the next section, we solve for \xx from the stock incentive problem of the manager. The worker makes his portfolio decision based on the distribution of q and his expectation about I. Since we are in a dynamic programming context, at t = 2, the worker learns the true level of q, i.e. q, and chooses his effort level consequently. Therefore, the incentive constraint yields the effort function Indeed, when q < 0 the effort level is zero. The expression (2.4.6) captures the option value to the worker, and it is this option feature which makes the worker hold more company stock than he would hold otherwise. The effort is proportional to the share-holdings. Al though we do not formally allow trading, at this stage of the game, we specu-late that allowing the worker to trade would not substantially change the equil ibrium. Indeed, suppose that the worker can trade his shares after observing his productivity characteristic q. We argue that the worker won't trade. It is important to note that this is not captured by the model, here we are just speculating on what would have been the equil ibrium outcome. The intuition is as follows. Suppose that we have only two states of the world: the high productivity q = qu and the low productivity q = 0. The worker can observe qH and decide to sell his shares at the high price and then decide not to work. However, that can't be the case because the market w i l l interpret this action of the worker as not wi l l ing to exert effort and the price w i l l adjust consequently. Thus it (2.4.6) is in the interest of the worker to exert the corresponding effort level after observing his productivity characteristic g#. The maximization (2.4.5) yields the following proposition. Proposition 2.4.1. The optimal allocation in company stock is A = (A*i ~ rf) - (Mm - r / ) p a 1 / a m  1 7 ^ ( ( 1 - P > 2 + 5 ^ W 2 ) - W 2 ' [ • • 1 the holdings in the market equity is Am = f ^ l - p ^ A 1 . (2.4.8) A\ increases with n2 over the range [0, fj2], and decreases over the set [fj2, +oo], where fj2 is a given threshold and can be obtained numerically. The compensation wo is such that the worker's participation constraint binds, i.e., E[Uw] = Uw, which implies w0 = (Uw - A1(n1 +Arf/A-rf) - Am(/j,m-rf) +0.57w[A2(a2 + Affi/A) + A2ma2m + 2AlAmPalam}) /'(1 + rf). (2.4.9) Proposit ion 2.4.1 has the following implications. First ly , consistent wi th basic intu -i t ion, if the firm has a lower risk, or a higher productivity, the investment in company stock w i l l be higher. From Equations (2.4.1) and (2.4.6), at t = 0 the unconditional variance of the f irm is G\ + A\rfhj\. The firm's volatil ity is affected by three elements: o\, A\ and r\. The conditional variance of the f irm taken at date 2 is a\. The risk level of the company affects the worker's holdings of company stock, A\, through two channels. O n one hand it affects directly the shareholdings of the worker. O n the other hand, the effort level of the manager (HI) changes wi th the risk level of the company, which affects the shareholdings of the worker as well. W h e n the conditional variance of the f irm is lower, the worker invests more in com-pany stock. Mitche l l and Utkus (2002) report that "... a recent survey of national D C plan participants showed that participants systematically err in assessing the risks of their company stock (Figure 1), rating employer stock as less risky than a diversified equity mutual fund. Moreover, that survey showed that participants properly rated ' individual stocks' as more risky than an equity mutual fund, but they considered their employer's stock as less risky (in effect they perceived their own company stock as less risky than other individual stocks). Despite the fact that average volati l ity of an individual stock is at least twice the volati l ity of a diversified market portfolio, participants rated individual stocks as only slightly more risky." In light of our analysis this assessment of company volati l ity seems rational. Indeed, when asked about the volatil ity of his company stock, the worker w i l l give the volatil ity conditional on the state he is in since he is insider to the firm. But the econometrician sees the unconditional volatil ity which is bigger than the conditional volatility. Secondly, if the worker is less able to influence the value of the f irm, the investment in company stock w i l l decrease because there isn't enough rent to be extracted from the company. In the expression of A\ (eq. 2.4.7), rj2 captures the option value of the worker exerting effort. Consequently, the worker's shareholdings of company stock increases wi th rj2 over the range [0, fj2], but when rj2 becomes large, the cost of lost of diversification dominates, and then the worker decreases his holdings in company stock. That can be seen from the expression of dAi/drj2 dA± ^ • Al{l/2-hlwA\rj2) On2 l w ((1 - p2)a\ + 15A ? 7 7 4 / 2 ) - n 2 / 2 " The investment in company stock A\ increases wi th the variable Indeed, from the first order condition ( F O C ) of the ut i l i ty maximization problem, the variation of A\ with respect to \i\ is: dAj _ 1 dm ~ jw ((1 - P2)o\ + 1 5 A V / 2 ) - « 2 / 2 ' In the next section, we show how fix is chosen by the manager. The empirical evidence documented by Huberman and Sengmueller (2004) suggests that workers rely on past return performance of the f irm to make their portfolio choice, and then they invest more in own company stock if the company stock has performed well over the past years. To reconcile this finding wi th our model, we first argue that HI is proxied by past returns performance, and since a high level of \i\ gives a signal to the worker that a high productivity manager is in charge, the worker's investment in company stock increases in order to benefit from the profit sharing. Next we solve the manager's incentive problem, and show how the shareholdings of the worker is linked to the productivity of the firm. 2.4.3 T h e manager's compensation and firm productivity In the previous section, we have assumed \i\ to be firm specific and exogenously given. It is revealed to the worker. In this section we show how /ii is chosen by the manager. A s stated earlier, at t = 0 shareholders hire the manager wi th the compensation package (b,s), where b is the pay-for-performance sensitivity (PPS) and s the monetary compensation. They expect the manager to put an effort [i\ to increase the productivity of the firm. From equation (2.4.7) in proposition 2.4.1, the manager's effort, /ii, influences workers investment in company stock. We use a linear compensation scheme. The manager's end-of-period wealth is WM = b(l + r1) + s, (2.4.10) where b is the P P S and s is the monetary compensation. Since shareholders cannot observe the manager's effort, they determine the compensation package of the manager based on their estimation of her effort fii and the worker's effort I.4 The manager bears a disutil ity for putt ing the effort Hi, represented by the cost function C M ( M I ) - The cost function C M ( M I ) is an increasing and convex function of fj,\: c'M{m) > o, c^ ( M l ) > o, fyi > o. The manager is risk averse wi th constant absolute risk aversion coefficient 7 M - Similar to the argument about the welfare function of the worker, since the returns are not nor-mally distributed, we use the Mean-Variance ut i l i ty function for the manager. However, 4 W e could easily allow the manager to trade the market, thus the manager will be able to diversify part of his portfolio. However, this extension does not add much to our main focus in this paper. this ut i l i ty representation is not equivalent to maximize the expected negative exponen-t ia l ut i l i ty function. Nevertheless, we use the Mean-Variance ut i l i ty function since we assume the manager likes returns and dislikes risk. Hence, the manager's expected total welfare is E[UM] = E[WM] - 0.57MVar(WM) - C M ( M I ) - ( 2 - 4 . 1 1 ) A t the beginning of the period, shareholders set the compensation package of the manager. They do so by maximizing their expected end-of-period wealth wi th respect to the P P S , b, and the monetary compensation, s, under the manager's participation con-straint (or reservation util ity) and incentive constraint, and workers' incentive constraint. A s commonly stated in the corporate finance literature, we assume outside sharehold-ers to be risk-neutral agents since they hold well-diversified portfolios. Shareholders maximization program is then formulated as follows: maxE[(l-b)(l + ri)-s], ( 2 . 4 . 1 2 ) b,s under the manager's participation constraint, E[UM] > UM, incentive constraint, fj,i = a r g m a x U M , and the worker's incentive constraint, I = axgmaxUw Proposit ion 2 . 4 . 2 summarizes the optimal policies. Proposition 2.4.2. The manager's PPS is (d^/8b)(l + ( a A 1 / 9 M l ) r ? 2 / 2 ) 7 M ( a 2 + 5 A 2 r ? 4 / 4 ) + (3A*I/0&)(1 + (0A1 ' her effort is ( 2 . 4 . 1 3 ) /X! = C'^ib + 6 ( a A 1 / a ^ 1 ) r ? 2 / 2 - ^Mb^dAJd^A^/A), ( 2 . 4 . 1 4 ) where C'jff1"1 is the inverse function of C'M, and her monetary compensation s is such that the participation constraint binds, i.e., E[UM] — UM-' s = UM-b(l + ii1 + A1ri2/2) + 0.5^Mb2(a21+A21ri45/4:) + CM(m). ( 2 . 4 . 1 5 ) Proposit ion 2.4.2 has several implications. O n one hand, the results are consistent wi th previous findings asserting that the P P S b is decreasing wi th the risk level of the f irm and the risk aversion coefficient of the manager. The effort of the manager, / J i , is increasing wi th the P P S , b over the range [0,77] and decreases when 77 > 77. O n the other hand, we find that the P P S decreases for large values of 77. The intuit ion is as follows. The parameter 77 represents the volati l ity of the degree to which the worker's productivity affects the firm's value. W h e n 77 is high, the volatil ity of the firm returns increases, hence shareholders grant less shares to the manager. Thus the stock compensation of the manager is less because of the high uncertainty surrounding her abil ity to influence the worker's involvement. Propositions 2.4.1 and 2.4.2 show the relationship between the manager's productivity and the worker's wealth allocation in company stock. 2.5 Conclusion In this paper we propose an incentive model in which the worker chooses voluntarily to invest his D C pension wealth in the shares of the company that employs h im. The essence of our argument is that the worker does this in order to benefit from the option embedded in his abil ity to increase productivity through his own effort. The pension plan holdings allow h i m to internalize some of the externalities that would otherwise be captured entirely by outside investors. In addition, we show how the employees' portfolio allocations are related to the pro-ductivity of senior management. Senior management is assumed to be able to influence the general productivity of the work place and hence the productivity of the employ-ees. A s a result, senior management shareholdings and the shareholdings of employees through their defined contribution pension plans are shown to be consistent and part of the overall optimal employment contract of the firm. Our explanations bui ld on the incentive and information properties of stock own-ership and hence our work relates to the standard principal/agent framework that is widely employed to study management compensation. This literature, however, relies on binding contracts to implement the optimal shareholdings and therefore cannot explain the investment allocations of D C pension plans since D C plans are not bound to invest in the shares of the employer. We, therefore, add to this literature by deriving an optimal contract when employees are allowed to either buy or sell the shares of the company that employs them. 2.6 Appendix Proof of Proposition 2.4.1 The first order condition of the incentive constraint of the worker implies: dUw(q)/dl = qA1 - C'W{1) = 0, which implies I = Ai max(q, 0). Given the incentive constraint condition, the worker's expected welfare at t = 0 is: E[UW] = E[Ww]-0.5ywVar(Ww)-E[Cw(l)] = i4i(jui + AiE[q2lq>0} - rf) + Am{p,m - rf) + w0(l + rf) -0 .5 7 , w A{{a{ + A2Var(q\>0)) + A2ma2m + 2A1Ampa1a71 -0.5AlE[q2lq>0}. Since E[q2lq>0] = rj2/2 and Var(q2lq>0) = ?745/4, maximizing this expected welfare yields the following first order conditions: dE[Uw]/dA1 = (/ij + V2A1 - rf) - 7tv(Ai((7? + 2A 27} 45/4) + Ampaiam) - AlV2/2 = 0, dE[Uw]/dAm = {fj,m - rf) - -fW{Amo2m + A^po^am) = 0. The results are obtained by solving simultaneously these two equations. Thus the in -vestments in risky stocks are A = (A*i ~ rf) ~ (Mm ~ rs)p0il<Jm 7 f f ( ( l - p 2 K + 5 ^ / 2 ) - 1 2 / 2 ' Am = r~-p—Ai-The differential of Ai w i th respect to rj2 is dAi Ai(l/2-5lwA2r)2) drj2 l w ((1 - p2)a2 + 15Afr? 4 /2) - r ? 2 / 2 ' and the differential of Ai w i th respect to p,i is dAi 1 dm ~ lw ((1 - P2)a2 + 15A 277 4/2) - v2/* Proof of Proposition 2.4.2 We use the incentive constraint condition of the worker I — A\ max(g, 0). The incen-tive constraint condition of the manager obtained from the F O C of the maximizing of his ut i l i ty function is given as follows , .rfdAi , 5r?4 . dAi . + 2~cM ~ 4 ~cM ~ M ^ = Wri t ing the lagrangian for the optimization (2.4.12), the maximization problem reduces to m a x ( l + M l + AlV2/2) - 0.5lMb2(a2 + A2V45/4) - C M ( M I ) , b with pb\ obtained from the manager's incentive constraint. The F O C of this maximization is dfix n2 dA± , 2 , „ 2 5 r ? \ h2 5 » 4 AdAldii1 -% + Jd^~db ~ 6 7 m ( C T i + - b l M ^ M W , ^ b ~ C M - d b = °-Using the incentive constraint expression of the manager given above, this F O C becomes ( 1 + -2^-db ' h l M ^ . + - 6 ( 1 + -2 8J7M = °' and the result follows. 3 Essay 2: Excessive Holdings and Equi l ibrium Asset Prices 3.1 Abstract In this paper, I study the equil ibrium implications when some investors in the econ-omy overweight a subset of stocks wi th in their portfolio. I find that the excess returns for the overweighted stocks are lower, al l else being equal. This has strong testable i m -plications for stock returns. In the special case of logarithmic preferences, the riskfree rate increases and the market price of risk for the overweighted stock decreases, which create extra incentive for unconstrained agents to exit the stock market and hold bonds, hence clearing the market. The changes of stocks' volatilities are ambiguous. Final ly , I provide an accurate quantification for agents' welfare. I also discuss the implications of my model in the context of defined contribution pension plans where workers hold large shares of their employer. 3.2 Introduction Common wisdom suggests that we diversify our portfolio holdings. However this advice is not followed by many investors. For example, it is reported that, more than a th i rd of defined contribution (DC) pension plans assets are invested in own company stock [Benartzi (2001), Benartzi and Thaler (2001), L iang and Weisbenner (2002), M e u l -broek (2002), Mitche l l and Utkus (2002), and Huberman and Sengmueller (2004)].1 This allocation runs contrary to standard investment advice. This extra non-diversification risk is further increased for workers because their human capital is already highly cor-related wi th their company stock. A t the end of 2000, D C plans had 2.5 tr i l l ion dollars total assets under their control, equivalent to about 20 percent of the total market capi-talization of the New York Stock Exchange [Huberman and Sengmueller (2004)]. Given •"•There are two basic types of pension plans: defined contribution (DC) and defined benefit (DB) plans. In D C plans (or 401 (k) accounts), the employer agrees to contribute a certain amount in each period to employees' pensions. The employees bear all investment risk since the firm has no obligation beyond making its periodic contributions. In D B plans, the employer promises to pay a certain amount of benefits at retirement, which depends on the employee's years of service and history of wages over the employment period. Risks are borne by the firm. the magnitude of D C plans assets, overinvesting in company stock wi l l have implications for equity prices and workers' wealth. In this paper I study the general equil ibrium implications of excessive holdings of stocks in a multiple-stock economy populated by heterogenous agents. I use the continu-ous time framework to ensure tractability. I consider two classes of agents wi th constant relative risk aversion preferences. The first class is composed of agents who can hold any admissible portfolio, and the second class consists of agents who invest excessively in a subset of stocks. I call agent in the first class, unconstrained, and in the second class, constrained. Thus agents in this economy are heterogeneous in their investment behaviour. I impose an exogenous holding floor on constrained agents, and use the convex-duality technique to solve for their portfolio choice problem. In order to derive the equil ibrium policies, I construct a representative agent wi th state-dependent preferences by assigning a stochastic weight to the class of constrained agents. The stochastic weight allows for the shifting of wealth from constrained agents to unconstrained agents, and it is key to the characterization of the equilibrium. I use the economy wi th no investment restrictions on agents (i.e., a dynamically complete economy) as the benchmark case. I character-ize explicitly the equil ibrium. I also provide approximate closed-form expressions for stock returns' volatilities using perturbation methods in the special case of logarithmic preferences. I find that the restricted economy deviates from the standard consumption-based C A P M , and failing to account for this deviation w i l l result in a higher cost of capital for the overweighted stocks. In other words, the excess returns for the overweighted stocks are lower relative to the benchmark al l else being equal, which has strong testable asset pricing implication for cross-sectional differences in expected stock returns. Indeed, when a positive shock occurs, the willingness of unconstrained agents to hold risky securities increases, but the residual supply of the stock held excessively is artificially constrained, hence the increase in its price. The volatil ity of the overweighted stock may increase or decrease depending on the init ia l parameters values of the dividends processes. W h e n agents have logarithmic preferences, the equil ibrium riskfree rate goes up and the market price of risk for the overweighted stock falls. The intuit ion is as follows. Constrained agents prefer a lower interest rate since they are net borrowers, while un-constrained agents require a higher interest rate to have an incentive to hold bonds. Therefore, the realized interest rate has to be high for unconstrained agents to clear the bond market. Moreover, the decrease of the market price of risk for the overweighted stock creates an extra incentive for unconstrained agents to exit the stock market and hold bonds. The direct consequence of the lower market price of risk for the overweighted stock and the high riskfree rate is that constrained agents bear idiosyncratic risk for which they are not compensated, leading to a welfare loss which is increasing in the degree to which holdings are constrained. The vulnerability of unconstrained agents' consumption to shocks is reduced in the restricted economy relative to the benchmark. I observe the opposite effect for constrained agents' consumption. The intuition for this difference in consumption volati l ity is as follows. In this economy we assume that the supply of bond is in zero net supply. Therefore the constrained agent is a net-borrower and the unconstrained agent a net-lender. B y holding more bonds than he w i l l otherwise do in the benchmark economy, the unconstrained investor benefits from the high riskfree rate and reduces his consumption volatility, especially to idiosyncratic shocks from the production process of the overweighted stock. Final ly , I quantify agents' welfare variation and show how welfare is transferred from the class of constrained agents to the class of unconstrained agents. M y welfare quantifi-cation takes into account the equilibrium price changes arriving from the distribution of wealth among agents, thus it is more accurate and can be used to quantify welfare loss in economies wi th under-diversified agents widely documented. 2 2Indeed, Brennan and Torous (1999), Meulbroek (2001, 2002), and K a h l , L i u and Longstaff (2003) estimate the cost of under-diversification to C E O s and investors. Since their frameworks are not in general equilibrium settings, they do not account for price changes resulting from these investments behaviour, thus the overall feedback effect on welfare quantification is missed. Dynamic equil ibrium models of optimal portfolio and consumption choice have re-ceived a lot of attention since the seminal work of Merton (1971). Cuoco (1997) pro-poses a partial equil ibrium model in the presence of portfolio constraints. His framework, however, has no implications for the risk-free rate and stock volatilities. He and Pages (1993) and Detemple and Serrat (1998) discuss the equilibrium implications for asset prices when agents face borrowing or l iquidity constraints preventing them from bor-rowing against their future income. Basak and Cuoco (1998) solve the equil ibrium in an economy where a class of agents faces information costs or other frictions preventing them from investing in the stock market (i-e., restricted stock market participation). De-temple and M u r t h y (1997) and Basak and Croitoru (2000) study the equil ibrium asset prices when agents have heterogeneous beliefs and face l imited short-sale and/or bor-rowing constraints. Merton (1987) and Shapiro (2002) develop models of capital market equil ibrium when investors buy and hold only stocks they are informed about. M y setting is closed to Shapiro (2002) even though we don't consider the same con-straint space. Indeed, in his setting, because of the lack of visibi l ity on some stocks, the class of constrained agents implements a particular trading strategy which reduces the opportunity set to a riskless asset and a managed fund. M y framework differs from his in at least two respects: first, his constraint space is unable to replicate mines, and second, the opportunity set of the class of constrained agents in my model cannot be synthesized to a risk-free security and an aggregate fund. Other models of equil ibrium asset pricing account for heterogenous beliefs and/or portfolio constraints wi th a single risky asset, e.g., Basak and Croitoru (2000), Heaton and Lucas (1996), Marcet and Singleton (1998), Kogan and U p p a l (2001) and Gallmeyer and Hollif ield (2002). M y setting also differs from the 'large' investor framework in which large institutional investors are able to move prices by their unilateral trading [Cuoco and Cvitanic (1998) and DeMarzo and Urosevic (2000)]. In my setting, instead, the economy is populated wi th a continuum of small investors, and some of them overinvest in particular stocks. One direct application of my framework is workers' D C pension plans. M y analysis shows that, in addition to Merton's (1987) factors pertaining to explain cross-sectional differences in stock returns (common factors, stock's relative size, idiosyncratic risk and investors' base), the diversification opportunities emerge as potential contenders for com-panies having D C pension plans. This result has strong testable implications. Indeed, Mitche l l and Utkus (2002) found empirically that excessive stock ownership in D C plans is a large firm phenomenon; Benartzi and Thaler (2001) and Liang and Weisbenner (2001) found the plan design, such as number of investment options offered, to be very important in determining the shares of 401 (k) assets in company stock. M y framework can also be used to analyze economies with undiversified C E O s or home bias investors. The rest of the paper is organized as follows. In Section 3.3, I describe the economy environment, and solve the unconstrained and constrained agents' optimization prob-lems. In Section 3.4, I characterize the equilibrium. In Section 3.5, I specialize my results to the special case of logarithmic preferences. In Section 3.6, I conclude. The proofs and the simulation procedure are presented in the Appendix . 3.3 T h e economic setting I consider a continuous time finite horizon [0,T] economy, wi th three securities: an instantaneous riskfree bond and two risky stocks. 3 I denote by B the bond price, Si stock l ' s price and S2 stock 2's price. 3.3.1 Securities market and information structure The uncertainty in this economy is generated by a two-dimensional Brownian motion Z = [Zy^Z-i)1 in a complete probability space (f2, JF, F , "P) where F = {Tt\ t G [0,T]} is the augmentation under the probability V by the nul l sets of the filtration generated by {Z(t); t € [0, T}} and T = T T - A l l stochastic processes in this paper are assumed to be adapted to F , and al l equalities involving random variables are understood to hold "P-almost surely. To simplify notations, I denote by Et[ ] the conditional expectation 3 T h e generalization of this framework to more than two risky assets case is straightforward. However, with only two risky stocks I can convey the main message of this paper without loss of generality. with respect to the available information set Tt at time t, E[ /Ft}- For any random variable x, I use x(t) in place of x(t,uj), where w e fi (is the state of nature). I use ||x|| to designate the norm of vector x. The two stocks represent claims to dividends 5\ and 52, which processes are given by d8j(t) = Sj^lns^dt + as^dZit)], j = 1,2. I denote by p the correlation between the two dividend processes. It w i l l be shown in equil ibrium that the stock prices follow an Ito process dSjif) + 5j{t)dt = Sj(t) [iij(t)dt + aj(t)TdZ(t)] , j = 1,2, and the bond price satisfies dB(t) = r(t)B(t)dt, where r, pj and o~j are adapted to F and are determined endogenously. I denote by p, — (pi,p2)T the instantaneous mean vector of stock returns, E = (a i , or 2 ) T the volati l ity matr ix and p 1 2 the correlation between the two stocks' returns. I w i l l assume for now that the matr ix £ is investible, and a l l processes r, £L, S , and S _ 1 are bounded uniformly on the set [0,T] x Cl. Since S is endogenous, whether it is of full rank is an equil ibrium property of the economy and can only be verified in equilibrium. For later use, I introduce the following process, d£(t) = -£(£) [r(t)dt + 6(t)JdZ{t)] , w i th f(0) = 1, (3.3.1) where 0 — T,~1(p — rl) w i th 1 = (1 ,1 ) T is the relative risk price. A s is well-understood in the literature, £(t,w) is interpreted as the Arrow-Debreu price per unit probability V of one unit of consumption good in state w € fi at time t G [0,T]. 3.3.2 Agents A s motivated in the introduction, the economy is populated by two classes of agents, indexed by 1 and 2. B o t h classes of agents consume part of their wealth and invest the rest in the financial market. They make their investment decision over the time horizon [0, T]. The only difference between the two classes of agents is that agents in class 2 are constrained in their holding of stocks, which w i l l be described shortly. A l l agents in the economy have preferences represented by intertemporally additive expected ut i l i ty w i th constant relative risk aversion ut i l i ty function rT U = E [ e-0tu{c{t))dt + e-0Tu(w(T)) Jo where u(x) = x 1 _ 7 / ( l — 7) w i th 7 ^ 1 and 7 > 0. W h e n 7 = 1, I have the l imit ing case of logarithmic ut i l i ty function. The quantity c(t) represents the consumption of the agent at time t and w(T) is his terminal wealth. 4 The parameter (5 represents the agent's inter-temporal discount rate. Agent i has an in i t ia l wealth i u j (O ) . Trading takes place continuously at the equil ibrium prices. A n admissible trading strategy is a two-dimensional vector process 7Tj = (niti, 7 r i j 2 ) T , where 7TJJ is the percentage of agent i 's wealth invested in stock j (j = 1,2). The remaining percentage of wealth, 1 — l T 7Tj, is invested in the bond. The portfolio weight vector 7Tj is assumed to satisfy the integrability condition f (||(1 - l T 7r l ( i ) ) r ( i )|| + | |7r i (t) T A(*)l l + |K(t) T E(i)|| 2 )dt < + 0 0 . Jo The portfolio weight vector 7Tj is said to finance the consumption plan c if it satisfies the budget constraint dw(t) = w(t) (TTi(t)T'(/2(t) - r ( i ) l ) + r(t)) dt - c(t)dt + w(t)TTi(t)TY,(t)dZ(t). I assume that agent 1 can hold any admissible portfolio while agent 2 can only hold portfolios, 7T2, such that 7r2,2(*) > 7T2, (3-3.2) where 7f2 is exogenously given. The constraint (3.3.2) is meant to capture in a reduced 4 I n the case of workers' D C pension plans, w(T) represents the pension total wealth at the target date T. form the excessive holdings of stocks observed in the real wor ld . 5 Using portfolio con-straints to model the restricted market participation is not unique to my model, Basak and Cuoco (1998) and Shapiro (2002) used the same approach. From now, I call agent 1 the unconstrained agent and agent 2 the constrained agent. Unconstrained agent's optimization problem The unconstrained agent faces a complete market, and as a result w i l l hold a well-diversified portfolio. Since unconstrained agent is facing no constraint, the unit state price he is facing is the economy state price density. A s in Cox and Huang (1989), the unconstrained agent's uti l i ty maximization problem can be stated as a variational problem wi th a single budget constraint, i.e., Uo J A t optimum, the marginal rate of substitution of one unit of consumption at state to and time t is proportional to the economy state price density, hence the optimal consumption and terminal wealth are: c*(£) = (e^£(i)y) h and w\(T) = (e^^(T)y) / 7 , where y is a constant positive given by y = (Wl(0)/E[ e-^^t^-^dt+e-^^T)^-1^})^. Constrained agent's optimization problem B y assumption, the constrained agent holds at least a percentage TXI of his wealth in stock 2 at time t. Because of this lower bound constraint, agent 2 faces an incomplete market. His portfolio choice problem is solved using the convex duality technique of 5 G o o d illustrative examples are workers' D C pension plans. Liang and Weisbenner (2002) report that among D C pension plans that offer company stock as an investment option, between 30 and 40 percent of total assets are held in company stock. Hewitt Associates (2000) reports that D C plans participants continue to favor three asset classes: company stock (40 percent of wealth invested), large US equity (23 percent of wealth invested) and Guaranteed Investment Contracts (12 percent of wealth invested). The excessive holdings of company stock can be an endogenous decision, my model abstracts from this generalization and uses an exogenous holding bound to capture the excessive holdings. max c,w under the budget constraint •T E Cvitanic and Karatzas (1992). Here are the steps to obtain the optimal policies for agent 2. Consider the constraint space facing agent 2 K = |7r2 = (7T2,i,7r2i2)T € M 2 , such that 7T2,2 > ^2} , where 7 ^ is the percentage of wealth agent 2 invests in stock j (j = 1,2). The holding floor n2(t) is a stochastic process adapted to F . I define the support function of — K as: w t ; / \ , Ts f -K2V2 if (u i ,v 2 ) € {0} x R+, V v = (vi,V2) e l , ip(y) = sup (—7T2 V) = < 7r2eK ^ + 0 0 otherwise, The effective domain of the support function is K = [v = {v1,v2)T G M 2 , tp(v) < + 0 0 } = {0} x E + . The incomplete market facing agent 2 is mirrored by a new fictitious complete market where the agent faces a new riskfree rate r+ip(v), and mean return for stocks fi+v+'ijjiy)!. In my setting, the support function of the constraint set is non nul l on its effective domain, that affects the interest rate and mean returns. It appears as if agent 2 faces a lower riskfree rate, r — 7fu2, and a higher market price of risk for stock 2, 9 + S _ 1 t> . The fictitious state price density, £"(£), facing this agent follows dC(t) = ~C(t) [(r(t) + ip(v{t)))dt + (8(t) + E-\t)v{t))TdZ(t)] , w i th £"(0) - 1. The constrained portfolio allocation problem becomes an unconstrained optimization problem in the new fictitious complete market, and can be stated as follows max E c,w under the budget constraint / e-^u(c(t))dt + e-^uiwiT)) Jo E •T C(t)c(t)dt + C(T)w(T) 0 w2(0). A s shown in Proposit ion 3.3.1 below, this dual setting implies similar formula for the optimal policies as in the complete market case, except that the state price density is changed to the minimax state price density, where v* is obtained from the dual minimization problem. Proposition 3.3.1. At optimum, the optimal consumption and terminal wealth of agent 2 are: c*2(t) = {e^C,'(t)yv*)~lh andw*2(T) = (ef*rC'(T)yv')-Vr', whereyv" is a constant positive given by yv' = ( w 2 ( 0 ) / £ [ J g V ^ r * { t Y ^ ^ d t + e-^^(T)^^])"1, and v* solves inf E \ [Tu(yvC(t),t)dt + u(y"C(T),T)] , with u(y,t) the convex conjugate of e~^lu(x) given by u{y,t) = max 1 > 0 (e" ' 3 f w(2; ) — yx). Having solved the individual agents' consumption-portfolio choice, in the next section I address the equil ibrium issues. 3.4 Equi l ibr ium in a pure-exchange economy B y assumption, the riskfree bond is in zero net supply and the risky assets are in positive supply of one unit each. Initially each agent in class i (i = 1, 2) is endowed wi th one share of stock i. The aggregate dividend, 5, is the sum of the dividends paid by the stocks <K*) = *i(*) + <*2(*). and its underlying process is denned by the exogenous process d5(t) = 5(t)[ps(t)dt + as{t)TdZ(t)] . (3.4.1) The state variables in this economy are the dividends. I determine endogenously the interest rate, r , market price of risk, 9, and stocks price dynamics in equil ibrium. The market equity satisfies the following process dS(t) + 5(t)dt = • S(t) [n(t)dt + a{t)TdZ(t)) , which w i l l be shown in equilibrium. Definition 3.4.1. A n equilibrium is a collection of {r, fi, £ } and optimal {c*, ir*, w*, i = 1,2}, such that, ir* finances {c*(t),w*(T), t e [0,T]}, and the con-sumption good and securities markets clear: + 4(t) = 5(t), wl(T) + w*2(T) = 6(T), < j ( * K ( t ) + = SM j € {1,2}, w*1(t) + w*2(t) = S1(t) + S2(t) = S(t). In order to analyze the equil ibrium I construct the representative agent ut i l i ty func-t ion as follows u(6,\)= max ^ - 7 ( l - 7 ) + A < 4 - 7 ( l - 7 ) , (3.4.2) Cl+C2=0 where A represents a weighting parameter. Solving (3.4.2) for the representative agent ut i l i ty leads to u(5, A) = ( l + A 1 / , 7 ) 7 <5 1 _ 7 /(1 — 7), and the optimal consumption policies are c\ = 5/(1 + \ l h ) and c*2 = 5X^/(1 + A 1 / 7 ) . To help understand the equilibrium implications in the restricted economy, I first characterize the equilibrium for the benchmark economy where both agents can hold any admissible portfolio. This benchmark economy w i l l help understand the market distortions introduced by the abnormal holdings of stock 2 by agent 2. 3.4.1 T h e benchmark economy - no excess holdings case The following proposition characterizes the equil ibrium in the benchmark economy. Proposition 3.4.2. In the benchmark economy, the equilibrium state price density, riskfree rate, r, market price of risk, 6, and risk premia, p, — rl, are £(t) = e-*(6(t)/mr, ( 3- 4-3) r(t) =P + lti5(t) - 7(1 + 7 ) I K ( i ) H 2 A (3.4.4) 0(t) = >y<r6(t), (3.4.5) 1 /dSdt) dS(t), . , „ t n t Agents' consumption growth processes are dd(t)/ci(t) = ns(t)dt + as(t)TdZ(t), i = 1,2. The interest rate is decreasing wi th the aggregate consumption growth volati l ity ||cr$|| and the precautionary savings of agents. 6 The market price of risk, 9, is proportional to the aggregate consumption growth volati l ity vector as- The intuit ion is as follows. W h e n the aggregate consumption is more volatile, agents demand more safe security for hedging, which induces them to lower their risky stocks holdings. In equil ibrium, the interest rate decreases, creating an extra incentive for agents to hold risky securities. The risk premia equal the aggregate absolute risk aversion times the covariance between stock returns and the aggregate consumption growth, which is the standard Consumption-based Cap i ta l Asset Pr i c ing Mode l ( C C A P M ) obtained by Breeden (1979) and Duffie and Zame (1989). Furthermore, agents' consumption growths are perfectly correlated wi th the aggregate consumption growth. W h e n the aggregate dividend process is geometric Brownian motion, the market equity price is therefore, the market equity volati l ity coincides wi th the aggregate consumption volat i l -ity, a{t) = os(t), and the market premium is \i — r = 7||cT<5||2. Intuitively, when the aggregate consumption is more volatile, agents use the stock market to hedge the uncer-tainties, thereby transmitt ing the high uncertainty of future consumptions into the stock market. The price of domestic stock j is 6Indeed, the aggregate absolute risk aversion is —ucc/uc = 7/(5 and the aggregate absolute prudence coefficient is —uccc/ucc = (1 +7)/<5. sit) 6(t) [ 1 - exp { - ( / ? - ( ! - 7 K + 7(1 ~ 7)IKI172) (T - t)} I / ? - ( l - 7 ) M * + 7 ( l - 7 ) I K I I 7 2 + exp { - (p - (1 - 7 ) ^ + 7(1 - 7 ) I N | 2 / 2 ) (T - t)} + e x p { - / 3 ( r - £ ) } ( 5 ( T ) / 5 ( £ ) ) - 7 < 5 j ( T ) , j ' = l , 2 . W h e n nsj, tr^ , and a$ are non stochastic, Sj is simplified as follows: Sj(t) = 53{t)[j\xV{j\-P + ^ . ( r ) - 7 ^ ( r ) + ^p^Mr)|| 2 - jaj (r)aSj(r))dT}dS + exp{j\-P + M < 5 , ( r ) - 7 / * ( r ) + ^ ^ p ^ M r ) ! ! 2 - 7 a J ( r ) a ^ ( r ) ) d r } " . I now turn to the case of the restricted economy. 3.4.2 T h e restricted economy - excessive holdings case In the restricted economy, the market is incomplete for agents in the second class. To solve for the equil ibrium, I employ the representative agent method introduced by Cuoco and He (1994)7 The aggregate ut i l i ty constructed in Equation (3.4.2) depends on the weighting parameter A. When the economy has no restrictions, A is constant [e.g., Karatzas , Lehoczky and Shreve (1990)]. But when the economy has restrictions, A is an adapted process to F , and is not necessarily constant. The parameter A is crucial to the distribution of wealth between agents. The equil ibrium can be fully characterized only if we can find the process of A. Assuming the equilibrium exists, Theorem 3.4.3 provides its characterization. Theorem 3.4.3. In equilibrium, the underlying process of the weighting A is d\(t) = A(i) [nx(t)dt + ax{t)TdZ(t)] , with A(0) = y/yv", (3.4.7) where fiX(t) = 7 c r ^ ) T a A ( i ) + \\ax(t)\\2/(l + \{t)^) - n2(t)v*2(t), ax(t) = v*2(t)^l(t)i2 with t2 = ( 0 ,1 ) T , and v2 > 0. The state price density is The riskfree rate is it) = /3 + 7 M , ( i ) - 7 ( l + 7)||a(5(t)||2/2 (l + 7)\(t)^\\v*2m-\t)L2\\2 \{tyhv2{t)v*2{t) 2 7 ( l + A ( i )V7 ) 2 + l + \{tyh • V- »> 7 I t has been used by Basak and Cuoco (1998), Basak and Croitoru (2000), Gallmeyer and Hollifield (2002), and Shapiro (2002) among others in economies with restrictions different from mines. The market price of risk is 6(t) = 7 ^ ( t ) -A ( i ) 1 ^ v*2(t)Z \ty2. (3.4.10) 1 + \{t)lh The risk premia are lii(t)-r(t) = -y~nCov( dSi(t ) dS(t) ) (3.4.11) to(t) - r(t) 5i(t) ' S(t) dS2(t) d5(t) ) 1 + A(t)V7 A ( i ) 1 / 7 «2*(t). (3.4.12) A l l the equil ibrium variables in Theorem 3.4.3 depend on the excessive holding con-straint (3.3.2) as characterized by v2. A s shown before, v2 is positive when the constraint binds, and equal to 0 otherwise. The explicit expression of v\ is provided in Section 3.5 where agents have logarithmic preferences. Theorem 3.4.3 identifies several important differences between the benchmark econ-omy and the current economy where some of the agents hold excessively the second stock. F irs t , the risk premium in (3.4.12) deviates from the standard C C A P M (3.4.6). Indeed, after controlling for the usual risk premium, the expected excess return on stock 1 is higher than that on stock 2, al l else being equal. Th is has a strong testable asset pricing implication for cross-sectional differences in expected stock returns. In the context of D C pension plans, since it is well-documented that people who work for a publicly traded firm that offers D C pension plans tend to over-invest in the stock of that company, Theorem 3.4.3 predicts that stocks of these firms earn a lower premium, al l else being equal. Second, the riskfree rate expression in (3.4.9) has two additional terms compared to the benchmark rate (3.4.4). These two additional terms are functions of A(t) and v2. Since A(i) is time-varying because of the incompleteness of the market, the riskfree rate is more volatile. For example, if the aggregate dividend follows geometric Brownian process, the riskfree rate in the benchmark economy is constant, while the riskfree in the current economy is time varying. The intuit ion behind the difference is as follows. The fourth term in (3.4.9) is a discount to counteract the extra precautionary savings and the fifth a premium as incentive to hold bonds, I w i l l refer to them respectively as discount and premium. A s A increases, the premium dominates the discount. Intuitively as the holding floor increases, more interest rate compensation should be given to unconstrained agent to hold bonds and exit the market for stock 2. In Section 3.5 where agents have logarithmic preferences, I show that the premium is always greater than the discount. T h i r d , the market price of risk in (3.4.10) is equal to the benchmark ratio (3.4.5) diminished by the factor axA1^+ A 1 ^ 7 ) . A careful exploration of o\ shows a reduction of the market price of risk for stock 2, but for stock 1, the deviation is ambiguously linked to the correlation between the two stocks. Agent 1 needs both stocks to constitute his diversified portfolio. Because of the over-investment by agent 2, the net-supply of stock 2 is artificially constrained for other agents, then to have the equilibrium realized on the financial market, the market price of risk for stock 2 goes down and the riskfree rate goes up simultaneously to create incentive for agent 1 to decrease his demand for stock 2 and invest more in the bond. In addition, the market price of risk for stock 1 adjusts by its correlation degree wi th stock 2. If the two stocks are positively correlated, agent 2 short sells stock 1 to offset part of his exposure, hence the market price of risk for stock 1 increases creating incentive for agent 1 to buy it and clear the market for stock 1. Conversely, if the two stocks are negatively correlated, agent 2 demands stock 1 to reduce his exposure, and then the market price of risk for stock 1 has to decrease for agent 1 to sell it to agent 2. Stock prices and volatilities are characterized by the following lemma. L e m m a 3.4.4. The price of stock j is where £ is the pricing kernel. I denote by D t Y ( s ) the Malliavin derivative of the variable Y(s) at time t.8 Assuming all the regularity conditions are satisfied for the Malliavin 81 characterize the stock volatility using the Malliavin derivative chain rule. The Malliavin derivative of a variable X denoted by D ( X measures how perturbing is X to an innovation in the Brownian motion vector at time t. The Malliavin calculus has recently been applied in finance by Ocone and Karatzas derivatives to exist, the volatility of stock j is given as follows: <Tj(t) = 6(t) + Et Et +-J f <J i(a)(3£(a)/&f)D t<J(>)ds + ^ ( T ) ( ^ ( T ) / ^ ) D t 5 ( T ) Et f^(sMs)ds + am(T) Et +-/^•(s)(3£(s) /a\)D t A(a)cte + 8j(T)(d£(T)/d\)'Dt\(T) Et The stock price is the expected discount value of its future dividends using the pricing kernel. Its returns' volatil ity has four terms. The first term is the volati l ity of the pricing kernel. The second term is the impact from the perturbation of the dividend process of the stock. The th i rd and fourth terms show how shocks on the aggregate consumption and the weight affect the stock volatil ity through the pricing kernel perturbation. One main difference between the volatil ity of the stocks in this restricted economy and the bench-market is the last term. Indeed, in the bench-market, the last term is equal to zero since A is constant. Below I provide approximate closed-form expressions for the stocks volatilities when agents have logarithmic preferences. 9 The excessive holding of stock 2 by agent 2, not only affects the price of stock 2, but also the other securities prices. Having multiple stocks wi th some held excessively and others not, is very crucial in my framework. Thus my economy framework contrasts wi th the settings in Heaton and Lucas (1996), Marcet and Singleton (1998), Basak and Croitoru (2000) and Kogan and U p p a l (2001) who consider a single risky stock. (1991), Detemple and Zapatero (1991), Serrat (2001) and Detemple, Garcia and Rindisbacher (2003) among others. We refer the interested reader to the monograph by Nualart (1995) and Oksendal (1997) for the existence conditions. 9 Stock prices and the weighting process solve simultaneously a system of forward-backward stochastic differential equations ( F B S D E ) . Except Basak and Cuoco (1998) who obtain explicit closed form solution for the stock price, others have to rely on numerical approach to obtain the stock price and volatility, e.g., Basak and Gallmeyer (2002). However the problem I face here is complex because is endogenous, and also am dealing with a multiple-stock economy instead of a single-stock economy as in Basak and Cuoco (1998) and Basak and Gallmeyer (2002). Fortunately I can derive approximate closed-form expressions for stock prices and volatilities in the next section for the special case where agents have logarithmic preferences. Though in the bench-market, agents' consumption growths have the same dynamic as the aggregate consumption growth, in the restricted economy, the dynamics are not the same due to the stochastic nature of the weighting. Corollary 3.4.5 gives the consumption path of each class of agents. Corollary 3.4.5. The processes underlying agents' consumption growths are dc*{t)/c*(t) = /j,c*(t)dt + o-c*(t)TdZ(t), z = l , 2 , with drift + 1 + \{tyh) V 6 { t ) + 2 7 ( l + A(t)V7) ) ' ' 7 ( l + A(t)V7) (1 + 7 ) / ^ t . ™ , ( l -A( t )^ )||a A ( t )|| 5 7 ( 1 + A(t) 1/7) a A W + 2 7 (1 + A(t )V7) j ' and volatility vector = (r,(t) + < y ( 1 + ^ ) 1 / T ) t ; 5 ( t ) S - 1 ( t K Because of the constraint space of agent 2, the two agents are facing different invest-ment opportunity sets, therefore, their marginal rates of substitution are different. They pick differently their consumptions. The volatil ity of each agent's consumption growth is proportional to the volatil ity of the pricing kernel he is facing. Shocks on stock 2 tend to have asymmetric effects on the consumptions of the agents, agent 2's consumption is more affected while agent l ' s consumption is less affected. Since agent 1 invests more in bonds, his consumption growth drift increases wi th the riskfree rate differential. In addition, the covariance between the weighting changes and the aggregate consumption growth affects negatively the drift of agent l ' s consumption growth. A s this covariance increases, agent 1 loses potential wealth for not holding more of stock 2. For agent 2, the riskfree rate differential affects the drift of his consumption growth negatively. The consumption growth's drift of agent 2 is augmented by the covariance between the weighting changes and the aggregate consumption growth. In order to better understand the derived formulae in Theorem 3.4.3, I analyze the special case of logarithmic preferences in Section 3.5 below. I provide expressions for stock returns' volatilities and agents' welfare. 3.5 Restricted economy with logarithmic preferences Agent's preference is represented by a time-additive log ut i l i ty (7 = 1). I define [x}+ = max(x, 0) to be the positive part of x. From the log ut i l i ty feature, the aggregate market equity value is S(t) = (l-(l-P)e-^)5(t)/f3. Corollary 3.5.1 characterizes the equil ibrium in the restricted economy. A n d Corollary 3.5.2 provides the expressions for stock prices and volatilities. 3.5.1 Returns and risk premia The equil ibrium is characterized by the following Corollary. Corollary 3.5.1. The process of the weighting, X, is dX(t) = ^ - i f t ) ^ ^ [*»(*) ~ S2(t)/S(t)]\T2(Z-\t))TdZ(t), (3.5.1) with A(0) = w2(0)/w1(0) and t2 = (0,1) T . The riskfree rate and market price of risk are r(t) = (3 + »s(t) - \\a5(t)f + . ^ g ^ ^® [*•,(*) - S2(t)/S(t)]+ , (3.5.2) m = a s { t ) ~ i i s - ^L i i 2 ~ g » ( * ) / s ( * ) ] + s ~ i ( * ) t 2 - ( 3- 5- 3 ) The risk premia are In the expression Tr2(t) — S2(t)/S(t) , the term S2(t)/S(t) represents the percentage of wealth the constrained agent would invest in stock 2 if there were no constraint. Thus this expression measures the gap between the 'no constraint' holdings and the holdings in the restricted economy by the constrained agent. In Equation (3.5.1), The weighting process is positively correlated to the risk source 2, and then A increases wi th positive idiosyncratic shocks on stock 2, and decreases wi th negative shocks. B u t , A is ambiguously affected by the risk source 1. Indeed, when the two risky stocks are perfectly correlated, the constraint has no effect. The effect of the correlation between the two stocks appears through the expression l/||S _ 1 (t)i2|| 2 = (1 — / ^ ( i ) ) ! ! 0 ^ ) ! ! 2 - W h e n H/O12I  ~ 1, the constraint has less or no impact, because the constrained agent can offset the constraint effect by taking short positions on stock 1 for positive correlations, and long positions on stock 1 for negative correlations. Riskfree rate and market price of risk The riskfree rate and risk premia are both affected by the investment behavior of agent 2. In Equat ion (3.5.2), the riskfree rate in the restricted economy is higher than the bench-market rate. The additional component of the riskfree rate is proportional to the size of stock 2 in the economy and the degree to which agent 2 is constrained. The intuit ion is that, since the constrained agent is a net-borrower, the riskfree rate has to be high enough to create incentive for the unconstrained agent to hold more of the bond. Unlike the bench-market case, even when the aggregate dividend growth is normally distributed w i t h constant mean n$ and instantaneous volati l ity as, the interest rate, r , and market price of risk, 0, are st i l l t ime-varying in the restricted economy, while they are constant in the bench-market. The over-investment adds a additional risk dimension to the risk-free rate. Figure 3.1 plots the trend of the equil ibrium riskfree rate as function of the investment horizon and the holding constraint floor levels. For this plot and the other plots to come, I use the simulation method described in the appendix. I use fj,g = 0.04, consistent wi th the findings of Gordon (2000). Indeed, from Gordon (2000), the growth rate of the U S output has been roughly 0.04 over the last three decades. I also use as = ^=(0.04,0.04) T which corresponds to an aggregate consumption growth volatil ity of 4%, and is the same number used by Basak and Cuoco (1998) and Basak and Gallmeyer (2002). For the dividend of stock 2, since it is well known that individual stocks are riskier than the market portfolio, I use ns2 = 0.05 and a&2 = (0.050,0.245) T , corresponding to a volati l ity level of 25%. From the parameters values of the aggregate dividend and stock 2's dividend, I can compute easily the parameters for stock l ' s dividend. In Figure 3.1, the riskfree rate is increasing wi th the holding floor. Risk premia and market premium Equations (3.5.4)-(3.5.5) show explicitly how the standard C C A P M is adjusted to accommodate the over-investment in stock 2. For positive shocks on the aggregate consumption, unconstrained agents willingness to hold risky stocks increases, and then they w i l l value highly stock 2 whose supply is artificially constrained because of the excessive holding by constrained investors. 1 0 According to Merton (1987), there are four parameters causing cross-sectional differences in equil ibrium expected excess returns: (i) the exposure level to common factors, (ii) the relative size of the stock, (iii) the stock-specific risk leVel, and (iv) the relative size of the stock's investors base. To identify these components in Equations (3.5.4)-(3.5.5), I consider the example of D C pension plans. The term cov(dSj/Sj,d5/6') captures the exposure level to common factors. The term A = w^/wl representing the relative wealth of agent 2 can be used as proxy for the relative stock size as well as for the stock's investors base. Empir i ca l support for this is the finding by Mitche l l and Utkus (2002) that excessive stock ownership in D C plans is 1 0 I n some D C pension plans, the employer gives its own shares as matched contribution and some employers even impose restrictions on the selling of those shares. Benartzi (2001) reports that roughly a third of D C pension plans buy shares on the open market, and the remaining two-third issue shares. In this situation, constrained agents don't have to buy necessarily shares on the open market to satisfy their constraint. Stock 2 appears then as a scarce security for unconstrained agents, they value it highly. However, if constrained agents were required to buy shares on the open market to satisfy their holding constraint, unconstrained agents wil l be willing to sell the stock and clear the market only if the price is high enough. In either cases, the price of stock 2 increases with the holding floor. a large firm phenomenon. The firm specific risk is captured by l / | | X ! _ 1 i 2 | | 2 . r_ -1 + In addition, the term 7T 2 ( i ) -S 2 ( t )/S(t ) can be used as proxy for the diversification opportunities in the fund, which is one key contribution of this paper. Indeed, L iang and Weisbenner (2001) found that the plan designs, such as number of investment alternatives and employer match in company stock, are very important in determining the shares of 401 (k) assets in company stock. A n d Benartzi and Thaler (2001) reports that investors follow a 1/n naive diversification strategy, i.e., invest evenly across the funds offered as investment option in the plan. The market equity premium is p{t)-r{t) = \\as(tW - | | s i ( ( ^ 2 | | 2 [*»(*) - S2(t)/S(t)]+. The market expected return, fi, is not affected by A, however the market premium decreases because of the effect of the riskfree rate r . Figure 3.2 plots the equil ibrium market equity premium as function of the investment horizon and the holding constraint floor levels. The market equity premium is decreasing wi th the level of the holding floor. Therefore, the economy cost of capital is decreasing as the level of the holding floor increases. Whi le w i th restricted stock market participation, the equity premium increases [e.g., Basak and Cuoco (1998)], in the context of excess stock holdings, the market equity premium decreases instead. Recently, Jagannathan, M c G r a t t a n and Scherbina (2001) found that U . S . equity premium has decreased over the last three decades. Meanwhile, from 1985 to 2001, U .S . private sector D C pension plans have increased in number from a total of 461,963 to 700,000 plans, and in terms of total assets, from 0.43 to 2 tr i l l ion dollars [estimates from Mitche l l and Utkus (2002)]. Future empirical studies can draw eventual l ink between excessive stock ownership in D C pension plans and the reduction of the equity premium observed over the last decades. 3.5.2 Stock prices and returns' volatilities In general even in this simplified case where agents have logarithmic preferences, it is not possible to solve and obtain explicit closed-form solutions for stock prices and volatilities. Nonetheless, I use perturbation methods [see Judd (1998)] to obtain ap-proximate closed-form solutions. To achieve that, I first derive the forward backward stochastic differential equation ( F B S D E ) of the stock prices. The volati l ity expressions are obtained by applying Ito's lemma to the price processes. Second, I use continuation methods to solve simultaneously for stock prices and volatilities. The expression of the variable y is written as the Taylor series expansion of e: y = yo + Y,n=iyntn + 0(eN+1), where 0(eN+1) means that | | y - 2 / 0 - E l i J / n e n | | / | e J V + 1 | < oo. I use the F B S D E satisfied by the price process to solve for the coefficients of the power terms. Once I obtain the parameter estimates, I use the system of simultaneous equations to obtain the volatil ity of the stocks. The details of these steps are exposed in the Appendix . For a discussion on the precision of the perturbation methods, I refer the interested reader to Zeidler (1986) and J u d d (1998). I use the superscript 'o' to spell the optimal policies in the benchmark economy. For example, S° and a° are the price and volatil ity of stock j in the benchmark economy, and S° is the volati l ity matr ix in the benchmark economy. W h e n ^/s = M<5j ~ M<5 ~~ aJ{a&j ~ as) is non stochastic, S?(t) = 6j(t) exp{//(—(3 + fiSj/siT^drjds + exp{J^(—(3 + HSils[r))dT}\. A n d if ns.,s = 0, then £?(*) = (1 - (1 - p)e-^T-t^)8j(t)/p. To solve for stock prices and volatilities, I use e = [tT2 - 5|/5] + . I also assume fis2/s(t) = 0. The approximate closed-form expressions for stock prices and volatilities are given in the following Corollary. Corollary 3.5.2. Stock prices are S°2{t) 2A(t) ( l + 0.5S , 2°(t)/5(i) - log(S2(t)/8(t))) Si(t) = S°(t) S2(t) = S°2(t) sm I K / ^ I I I E O - H ^ ) ! ! 2 'ArfVm 1 + 0(e2) + 0(e2 2A(t) ( l + 0.5S^t)/8(t) - logW)/5(t)))' I K ^ W P U E o - 1 ^ ) ! ! 2 The volatility matrix £ = (<7i, a2)T is given by ax(t) S°2(t) 2A(t) ( l + 0.5S°2(t)/8(t) - log(82(t)/5(t))) SKt) ||a 5 2 /,(i)||2||S-Ht) t 2)|| 2 {(i&2(t) - aSl(t))e a2(t) The price of stock 2 increases, and then because of the equil ibrium property that of stock 1 decreases. Intuitively, since agent 2 is constrained to hold more shares of stock 2, the supply of stock 2 is artificially constrained and there is more demand for stock 2, leading to an increase in its price. But the changes of stock volatilities are ambiguous. 3.5.3 Consumptions and welfare Overinvesting in stock 2 affects the consumption paths and welfare of the two agents. The allocation of wealth amongst agents is affected through the weighting process A. A s stated in Corollary 3.4.5 and plotted in Figure 3.3, the two agents have different consumption paths in the restricted economy. Furthermore, in this special case of loga-rithmic preferences, the weighting process represents the relative consumption of agent 2 vis-a-vis agent 1: A(i) = c2(t)/c\(t). There is a suboptimal risk sharing in the sense that agent 2 bears more risk than agent 1 (Corollary 3.4.5), and the gap increases wi th the level of the holding floor, 7r2. W h e n ff2 is big, the idiosyncratic risk is born essentially by agent 2. A s Equat ion (3.5.3) shows, he is not compensated for bearing the excessive risk. Because of the excessive holding by agent 2, agent 1 increases his investment in the safe security, which smooths his consumption. The riskfree rate is then pro-cyclical and the risk premium is anti-cyclical. Positive shocks on stock 2 increase its price, which induces agent 2 to borrow in order to satisfy the constraint (3.3.2), consequently, the riskfree rate should go up to create an incentive for agent 1 to lend. The holding of bonds and the high riskfree rate concur to smooth agent l ' s consumption. The overall implication of differences in consumption growth is the change in the welfare distribution among agents. The welfare expressions are given below in L e m m a 3 . 5 . 3 . L e m m a 3.5.3. The welfare of the agent can be decomposed into two separate parts as follows: Ui = U° + AUi, where U° = E ' f f f> 4 (r) - \\o-s(T)\\2/2)dTdt + e-<3T C ( w ( r ) - \\O5{T)\\2/2)dr lJo Jo Jo J - log( ( l - (1 - P)e-^r)/Pwi(0))(l - (1 - (3)e-eT)/P, and A C / i = -E 2 +e -0T A 2 ( r ) 7f 2 (r ) - S2(T)/S(T) drdt dr AUo J T - J C ^ { [ ^ - ^ ] + } drdt +e" ^ / - T ( l + 2A(r)) 7f 2 (r ) - S2(T)/S(T) dr The first part, U°, is similar to the welfare in the benchmark economy, except that Wi{0) changes under the constraint and captures the wealth effect. The second part, AUi, is the gain/loss from the change in the hedging demand. In the economy setting, since agent 2 is endowed init ial ly wi th al l the supply of domestic stock, and wi th the increase of the price of stock 2 , the ini t ia l wealth of agent 2 in the restricted economy is higher than its benchmark level. However, the drift of the changes in the relative proportion of agent 2's wealth in the economy is negative: Et [d{X(t)/(l + A(£))}] = Et [d{w2(t)/(wi(t) + u>2(*))}] < 0- Thus agent 2's wealth is decreasing, which implies that the positive wealth effect disappears over time. From the expressions of AUi, we observe that agent l ' s welfare increases and agent 2's welfare decreases, which is consistent wi th our basic intuit ion. However, the novelty of L e m m a 3 . 5 . 3 is that it provides an accurate quantification of the welfare variation in the context of excess holdings. In that respect, it contrasts wi th the non-equilibrium welfare estimation in Brennan and Tofous (1999), Meulbroek (2001, 2002), K a h l , L i u and Longstaff (2003). Indeed, the existing literature doesn't account for equil ibrium prices changes caused by the undiversification behaviour of the agents. Hence the overall implication of prices on wealth distribution is over or under estimated. The stochastic weight A placed on agent 2 by the representative agent acts as proxy for stochastic shift in the distribution of wealth between the two agents. Prom the process of A in (3.5.1) and the expression l/||S _ 1 i2| | 2 , a high volati l ity for stock 2, and/or a low correlation between the two stocks, result in a huge welfare loss for agent 2 and significant gain for agent 1. Figure 3.4 displays the wealth shift from agent 2 to agent 1 in the restricted economy for different levels of holding floor. W i t h respect to our baseline values, there is a social welfare destruction even though the total wealth, w"[ + u ^ , is unchanged, al l else being equal. W i t h a holding floor of 30 percent, the average welfare loss for constrained agent is -1.04 percent and the average welfare gain for unconstrained agent is 0.66 percent. Doubl ing the holding floor to 60 percent, the average loss for constrained agent becomes -8.08 percent and the average gain for unconstrained agent 3.82 percent. The net social welfare loss is -4.26 percent. Holding an abnormal percentage of stocks exposes agents to idiosyncratic risk for which they are not compensated. These results parallel the findings in the context of workers D C pension plans. Meulbroek (2002) finds that, on average employees sacrifice 42 percent of the company stock's market value for lack of diversification. Equivalently, Brennan and Torous (1999) estimate the cost of non-diversification to be very high. 3.6 Conclusion In this paper, I study the equilibrium implications when some agents in the economy overinvest in a subset of stocks. I consider a continuous time pure exchange economy populated by two classes of agents. Agents are heterogenous with respect to their invest-ment behaviour. One group of agents is unconstrained, can hold any admissible portfolio, while the other group of constrained investors overloads in a subset of stocks. I solve for the equil ibrium by constructing a representative agent wi th state dependent weights assigned on the investors' utility. I find that the standard C C A P M is reduced by a factor proportional to the degree to which stocks are overweighted. Thus, the excess returns on the overweighted stocks are lowered al l else being equal. This adjusted C C A P M has testable implications for stock returns. W h e n agents have logarithmic preferences, the riskfree rate increases and the market price of risk for the stock held excessively decreases, making unconstrained agents exit the stock market and hold bonds. The consumption volatil ity of the un-constrained agent decreases and that of the constrained agent increases. I provide an accurate equil ibrium quantification for the distribution of wealth between agents. I also compute approximate closed-form solutions for stock returns' volatilities using pertur-bation methods when agents have logarithmic preferences. Since it is well documented that workers overinvest in employer shares wi th in their D C pension plans, I discuss my model implications in that context. To my knowledge, this is the first attempt to study the equilibrium implications of excessive holdings of particular stocks in a multiple-stock economy. Future extensions of my framework w i l l be to account for borrowing or l iquidity constraints, and also to include a wage income with an additional idiosyncratic source of risk. 3.7 Appendix Proof of Proposition 3.3.1 See Cvitanic and Karatzas (1992). Proof of Proposition 3.4.2 The representative agent ut i l i ty is constructed as: u(5, A) = max c l " 7 ( l - 7) + A c 2 " 7 / ( 1 - 7 ) . C\+C2=0 The F O C of this maximization problem gives: Ci/(S — Ci) — A - 1 / 7 , which implies: C\ = 5X^/(1 + A - J / 7 ) a n d c 2 = 5 - cx = 5/(1 + A " 1 / 7 ) . Substituting back in the ut i l i ty expression yields: u(5,X) = ( l + A 1 / 7 ) 7 5 1 _ 7 / ( 1 — 7). Lets denote by uc the marginal ut i l i ty of consumption of the representative agent. In the benchmark economy case, A is constant. The equilibrium state price density is given by: = e~0tuc(6(t), X)/uc(5(0), A) = (<5(£)/5(0))~7. App ly ing Ito's for-mula to gives de(t) = ((3 + 7 W ( i ) - 7 ( 1 + 7 )|M*)|| 2 /2 ) d i - £ ( i ) 7 ^ ( i ) T d Z ( i ) . Comparing this to the state price density expression given in Equation (3.3.1), gives the equil ibrium interest rate, r(t), and market price of risk, 9(t). The risk premia are ob-tained by p — r = HO. Proof of Theorem 3.4.3 From the individual optimization problems, e~^tu'1(c\(t)) — y £ ( t ) and e'^u'^c^t)) = yvC(t), where d£(t) = -£(t)[r(t)dt + 6(t)TdZ(t)} and (i) = -£" ( t ) [ ( r ( t ) -Tt2(t)v2(t))dt + (0(t) + S - 1 ( i ) w ( i ) ) T d Z ( i ) ] . Since agent 2 has a market restriction, the weight A assigned to the constrained agent is a measurable process wi th respect to the filtration F . From the F O C of the aggregate ut i l i ty construction, X(t) = u'x(ci(t))/u'2(c2(t)) = y£(t)/yv£v(t). App ly ing Ito's lemma to this expression of A yields dX(t) — X(t) [/i\(t)dt + a\(t)TdZ(t)]. The economy state price density is given by £(£) = e _ / J t i t c (5(£) , X(t))/uc(5(Q), A(0)). App ly ing Ito's formula to this expression of £(£), and substituting the expressions of dX(t) and d5(t) give the process for d£(t). The equilibrium riskfree rate r and market price of risk 9 are obtained by comparing the process of d£ (£) obtained here and Equat ion (3.3.1). The risk premia are obtained by p. — r — T,9. Proof of lemma 3.4.4 Stock prices are obtained by discounting their future dividends using the economy pricing kernel, £, Sj(t) = Et We denote by D 4 the Mal l iav in derivative operator at time t. Assuming a l l the regularity conditions satisfied for the different Mal l iav in derivatives to exist, we first compute the following Mal l iav in derivatives using Mal l iav in chain rule: D t £(s ) = ^ D ( A ( S ) + ^-Bt8(s) and D t £( i ) = - 0 ( t ) £ ( t ) . 86 We now take the Mal l i av in derivative of Si BtSj(t) D t g ( * ) +W)Et Et J\(s)sj(s)ds+amm Ut T J DtiZWjWds + BtMTMT)) + (5j(T)DtaT)+£(T)DMT)) 0(t)Sj(t) + ^ E t jT\( s )V> t ^s )ds + £(T)D t<$j(T) +^Et^\j(s)^Bt\(s)ds + 6j(T)^Bt\(T) J*8j(s)^-Bt5(s)ds + 8j(T)^-Bt6(T) Comparing this Mal l i av in derivative to the following Mal l iav in derivative of Sj, BtSj(t) = l i m D s 5 j ( £ ) = S ^ i ) ^ ) , gives Uj(t). Proof of Corollary 3.4.5 The proof follows by applying Ito's formula to the expressions of Ci(t) = 5(t)/(l + \{tfh) and c 2(t) = 5{t)\{tfh/{l + \{tfh). Proof of Corollary 3.5.1 The unconstrained financial market facing the constrained agent is: dBv(t) = (r(t) + ^{v(t)))Bv(t)dt, dS](t) + 5j{t)dt = S](t) [(nj(t) + Vj(t) + *p(v(t)))dt + o-j(t)TdZ(t)] , j = 1,2, w i th v = ( u i , u 2 ) T € {0} x K + and the support function ip(v) — —TT2V2. Using u(x) = log(x), the convex conjugate of e~^'u is u(y) = max I > o[e~ ' "u (3 ; ) — xy] = — e - / 3 t ( l + f3t + log(y)). The dual problem value is then: J{y,v) = E f u(yC(t))dt + u(y^(T)) Jo I need to solve minu G{o}xR+ J{y,v) to obtain v*. This is equivalent to minimize Et j T (r(s) + ^(v(s)) + \\6(s) + E- 1 (s) U (s)|| 2 /2) ds + ( r (T ) + *{v{T)) + \\9(T) + X-\T)v(T)\\2/2) Therefore, v*(t) = a r g m i n v e { 0 } x R + {-2it2(t)v2{t) + + E - 1 ( * M t ) | | 2 } . Solving this equation yields vl(t) = 0 and v%{t) = n2(t) - 6TT,-1(t)i2 / ||E- 1 ( i ) i 2 || 2 , w i th t2 = (0 ,1 ) T . Using the expression of 6 = as — v2 A K _ 1 t 2 , and after some algebraic manipulations, we obtain 1 + A(i) 7f 2 (i) - as(t) ' E - \ t ) c 2 I I E - ^ M I 2 Since we are in a log preference setting, we can show that the aggregate market equity value is: S(t) = (1 - (1 — P)e~^T~^)6(t)//3, hence its volatil ity is as- B u t since S = S\ + S2, its volati l ity is also the weight average of the two volatilities as follows: a5. = {S1/S)a1 + (S2/S)a2. Also recall E = ( C T 1 , G T 2 ) t So a]Z~h2 = S2/S, and v*2 is simplified as follows v*(t) = 1 + X(t) 7f 2(t) 52(£) S( i ) (3.7.1) , | s - i ( t ) t 2 | p W i t h this expression of v 2 and results from Theorem 3.4.3, the weighting process A is characterized by: a\(t) = E _ 1 ( t )v (£ ) and px(t) = 0. The riskfree rate r , market price of risk 9 and risk premia p, — r expressions follow. Proof of Corollary 3.5.2 To obtain the approximate stock prices and returns volatilities, I use perturbation methods. I use two steps to achieve that. The first step consists of computing the prices Si and S2. In the second step, I use the prices computed in step 1 to solve for the system of equations for the volatilities. The price of stock 2 is obtained by discounting its future dividends using the economy pricing kernel, S2{t) = Et 1 t W)2{) + W ) H ) . s°2(t) + ^ ) ^ H { m M t i m i 2 ) 1 + \{t) 1 + A(t) where S2 is the price of stock 2 in the benchmark economy, a(t) = 52(t)/5(t) and H(X(t),a(t),t) = Et e-f}sX(s)a(s)ds + e-'}T\(T)a(T) . Under appropriate regular-ity conditions, it can be shown that £ ( t ) S 2 ( t ) + f\(s)S2(s)ds is a martingale under V [see Cuoco and He (1994), and Basak and Gallmeyer (2002)]. Therefore, the drift must be zero. Hence, H solves the quasi-linear partial differential equation (c + H(X, a, t) + e-^X{t)a{t) = 0, (3.7.3) wi th boundary condition: H{X(T),a(T),T) = e~^TX{T)a{T), where £H(\,a,t) = ±HX\X2\\ax\\2 + i # a Q a 2 | | c 7 a | | 2 + HXaXaaJaa + HxXpx + Eaa\xa. I can not solve explicitly the P D E in Equation (3.7.3), but I can use perturbation methods to obtain approximate closed-form solutions. I do that as follows. I take the Taylor expansion of H w i th respect to a parameter e: H = H° + Z%=lHnen + 0{eN+1), (3.7.4) where Hn = f ^ ( e = 0) for n = 1,.. . ,JV and H° = H(e = 0). W h e n e = 0, I have the bench-market case. I choose e = [TT2 — S2/S~\ + . A t e = 0, S2 = S2, and then H°(\{t),a(t),t) = e-^X(t)S^(t)/6(t). To obtain Hns I have to solve the following P D E \\\a\fX2 mxen + \\\aa\\W ] T Hnaaen + aTxaaXa £ HnXaen + ^ Hnaen = 0. The price of stock 1 is obtained from the equilibrium condition as follows: S\(t) = s(t) - s2(t) = ( i - ( i - fle-w-vntyp - S2(t). To obtain the volati l ity expressions for the two stocks, I apply Ito's L e m m a to their price expressions, which yield '«) = *#) + ( 1 _ ( 1 _ p)J{l%{t)/p _ m (*( ' ) ~ *>(*)). (^.5) 1 r S°2(t) (e^5(t)Hx(X,a,t) S2(t) + i + \(t) a s { t ) + TTW) a{ } J • ( } I solve simultaneously for Hns and the volatilities ax, G\ and a2. Proof of L e m m a 3.5.3 Solving the agents' optimization problems give the following optimal policies. For agent 1: c\(t) = e^/^V, w*i(t) = e ^ l - (1 - Py-?^)/m)y, w i th y = (1 -(1 - /3)e- / 3 T ) / /?w 1 (0) ; A n d for agent 2: c*2(t) = e~^/^{t)yv*, io|(t) = e " ^ ( l - (1 -P)e-nT-Q)/P?*{t)y«', w i th j , " * = (1 - (1 - P)e^T)/Pw2(0). I use the expression of r and 6 in Corollary 3.5.1 and Equation (3.7.1) to obtain the explicit expression of the pricing kernels £ and £ w *. I use the expressions of c* and w* in the ut i l i ty function of agent i, and the result follows. Simulation procedure Lets denote by h the time interval, and then iV = ^ is the number of intervals on the axis [0, T] . I use the Euler scheme to simulate the paths of the weighting process A and the stocks dividends 5i,i E 1,2: X(t + h) = A(i) + \(t)ax(t)TAZ, w i th A(0) = w 2(0)/u>i(0), 5i(t + h) = 5 i ( i )exp(( M < 5 i (£) -||a, i (£)|| 2 /2) / l + c T 5 i ( i ) T A Z ) , i = 1,2, where AZ = (AZi, AZ2)r is a two dimensional gaussian vector wi th mean 0 and covari-ance matrix ( \. The aggregate dividend 6(t) = 6i(t) + 52(t). ^ 0 h ' The welfare in the restricted economy for the two agents are obtained by approxi-mating numerically the following expectation: U(t) = Et ^ log(C i( S))<2 S + e - « T - ' ) l o g K ( T ) ) The approximation of this expression is done in two steps. The first step consists of approximating the integral term. In the second step, the expectation is approximated. To approximate the following integral f(t)dt numerically, I use the trapezoidal rule: JV-l Q(f) = h[f (x0)/2 + J2 f(xj) + f(xN)/2 3 = 1 where ./V is the number of intervals on the axis [a, b], xo = a, xN = b, and Xj are the internal nodes elements. I then calculate the number inside the expectation by repeating the same procedure M times (which constitutes the M batches). I finally take the sample average of this M batches as an approximation of the expectation. Figure 3.1: E q u i l i b r i u m r i sk f ree r a t e . Interest rate 1 This graph plots the economy riskfree rate for different values of holding constraint floor 1T2 and investment horizon T — t (years). The baseline parameter values are [is — 0.040, as = -7s(0.040,0.040)T, ns2 = 0.050, as2 = (0.050,0.245)T, 0 = 0, 52(0) = 0.55(0). Figure 3.2: M a r k e t e q u i t y p r e m i u m . This graph shows the market equity premium for different values of holding constraint floor ff2 and investment horizon T — t (years). The baseline parameter values are fi$ — 0.040, as = -7= (0.040,0.040)T, fi52 = 0.050, a52 = (0.050,0.245)T, 0 = 0, 52(0) = 0.55(0). Figure 3 . 3 : C o n s u m p t i o n p a t h s . ConaumpUon change of agent 1 Consumption change of agent 2 These graphs show the percentage changes of consumption from the benchmark for the two agents for different values of holding constraint floor % and investment horizon T — t (years). L H S : agent l ' s consumption percentage changes. RHS: agent 2's consumption percentage changes. The baseline parameter values are ^ = 0.040, as = 4 (0.040,0.040)T, = 0.050, a52 = (0.050,0.245)T, /3 = 0, 52(0) = 0.5(5(0). Figure 3 .4 : W e l f a r e v a r i a t i o n . Welfare change I -0 .05 --0.1 -~ ° ' i 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Holding floor This graph shows the percentage changes of welfare for the two agents for different values of holding constraint floor 7f2 and an investment horizon of 40 (years), the dashed line is for agent 2 and the solid line for agent 1. The baseline parameter values are ft& = 0.040, o~$ = -4(0.040,0.040) T, tih = 0.050, = (0.050,0.245)T, 0 = 0, <52(0) = 0.55(0). 4 Essay 3: International Diversification, Investment Restrictions and T h e Exchange Rate 4.1 Abstract In this paper we develop a dynamic general equil ibrium model for a two-country, two-good exchange economy in the presence of investment restrictions. W h e n a domestic country caps foreign investment in some key industries in the domestic economy, the cost of capital of the protected industry increases, that of the non-protected industry decreases, al l else being equal. O n the other hand, when imposing restrictions on its residents' foreign investments, the domestic country improves its cost of capital, al l else being equal. Furthermore, in both restricted economies, the domestic cost of risk free borrowing and lending is lowered. However, when domestic residents are capped in their foreign investment, the uncovered interest rate parity relationship is violated. B y artificially restricting agents' investment, countries can reduce financial contagion effects because stock markets are affected asymmetrically. This result contributes to the debate on why recent crises in international financial markets have had different effects on countries located in same geographical area or having similar economic characteristics. The effects of the restrictions on stock market volatilities are ambiguous. Final ly , we show that when the restriction is protective, the welfare of the agents of the country imposing the restriction increases. This result helps us understand why some countries are so reluctant to change their protective financial policies. 4.2 Introduction The virtues of market liberalization have been lauded extensively in the international finance literature, in Dumas and U p p a l (2001), Henry (2003), and many others. 1 B u t x Dumas and Uppal (2001) affirm "...the welfare gain from integration of financial markets is not greatly reduced by the presence of goods market imperfections, modelled as a cost of transferring goods from one country to the other... The policy implication to be drawn is that financial market integration is a worthwhile goal to pursue even when full goods mobility has not been achieved." Henry (2003): "... Since the cost of capital falls, investment booms, and the growth rate of output per worker increases when countries liberalize the stock market, the increasingly popular view that capital account liberalization brings no real benefits seems untenable." despite the many claimed advantages, we st i l l observe governments that are reluctant to fully open their domestic capital markets even when they are well-developed. It is common to see countries artificially restrict their residents' investments abroad or l imit foreign investments in some key sectors of the domestic economy. The view that market liberalization is a worthwhile goal to pursue at any cost is not shared by eminent economists including the 2001 Nobel Prize winner in Economics, Joseph E . Stiglitz, who believes that market liberalization creates instability, increases poverty and economic insecurity in the liberalizing country. 2 In this paper, we develop a general equil ibrium model of international capital markets in the presence of investment restrictions, and study the implications on firms' financing and agents' welfare. We consider a continuous time, two-country, two-good exchange economy. Using the fully integrated financial markets wi th no investment restrictions as a benchmark, we consider two cases wi th investment restrictions. The first case is one where the domestic government caps foreign investment in some key industries in the domestic economy. This can be the case where the country seeks to protect some sensitive sectors of its economy. The restriction then prevents foreign investors from taking control of some key production processes, and also reduces the sensitivity of the domestic market to foreign investors' wealth variations. Examples include the airline and banking industries in Canada. The second case is when the domestic government restricts its residents' foreign investments. The "rationale" may be to hold capital inside the country, or to reduce the country's exposure to international shocks. Examples include Canadian pension fund investment regulations, which stipulate that Canadian pension funds cannot invest more than 30 percent of their assets under management in 2 Joseph E . Stiglitz in the September 22, 2003 issue of Time magazine criticized the International Monetary Fund (IMF) policies on globalization. According to him: "In fact, studies at the World Bank showed that capital market liberalization was systematically associated with instability, which increased poverty and economic insecurity and was bad for growth. The I M F had tried to argue that, without capital market liberalization, one could not entice foreign investors. Yet China had attracted more foreign direct investment than any other developing country, and it has stil l not fully liberalized its capital market. B y the end of the 1990s, it was hard to resist the mounting evidence... Finally, to its credit, the I M F took note of what had long been obvious: capital market liberalization does not enhance growth, and it does increase instability." foreign stocks. Several interesting results emerge from our study. First ly , when the domestic country caps foreign investment in some key industries in the domestic economy, the cost of capital of the protected industry increases, and then through equilibrium, that of the non-protected industry decreases, al l else being equal. O n the other hand, when imposing restrictions on its residents' foreign investments, the domestic country improves its cost of capital , al l else being equal. Furthermore, i n both restricted economies, the cost of risk free borrowing and lending is lowered in the domestic country. However, the uncovered interest rate parity establishing the relationship between foreign and domestic risk free rates v ia the exchange rate st i l l holds in the economy where foreign investors are restricted to invest in domestic protected industries. B u t , the relationship is not satisfied in the economy where domestic residents are restricted in the amount of their wealth they can invest in foreign capital market. In that case, the risk free rate in the foreign country increases relative to its benchmark level. Secondly, when agents' investment is restricted, stock markets are affected asymmet-rically by international shocks, which can lead to the attenuation of the international financial contagion effects. Our results contribute to the debate on why recent crises in international financial markets have had different effects on countries located in same ge-ographical area or having similar economic characteristics. A s stated by Joseph Stiglitz in T ime (2003), "... Every major emerging market that had liberalized its capital market had had a crisis; the two major countries that had not, C h i n a and India, had not only avoided the East As ian crisis, but managed to grow steadily throughout the period." Pavlova and Rigobon (2003) in a two-country, two-good exchange economy found that equity markets are perfectly correlated in equilibrium. Cass and Pavlova (2003) show that this perfect correlation st i l l holds even when investment restrictions are imposed. In their settings, each country specializes in the production of one particular good, thus the exchange rate is the channel of contagion from one stock market to the other. In our economy, instead, the firm in one country can produce the good available in the other country as well. This feature of our economy is more closed to what is observed in international markets, since it is common practice to see countries producing many commodities w i t h some available in other countries. W i t h this modelling feature, even in dynamically complete financial markets the correlation between the two equity markets is less than unity. Thirdly , when restrictions on investments are imposed, their effects on stock market volatilities are ambiguous. This result contributes to another debate in the international economics and finance literature and in financial market development related policies, that is, whether financial liberalization leads to more or less stable financial market. In particular, our result suggests that the debate on whether opening up financial markets is the cause of financial instability would not be fruitful at the general level because both more stable and less stable financial markets can occur in equil ibrium. Final ly , when restrictions on investments are imposed, depending on the nature of the constraint, the welfare of the agents of the country imposing the constraint can either increase or decrease. In fact, w i th log uti l ity, we show that when the restriction is protective, the welfare of the agents of the country imposing the restriction increases. Th is result helps us understand why some countries are so reluctant to change their protective financial policies. Our paper relates to a number of papers in international economics and finance l itera-ture. Zapatero (1995) determines the dynamic of the real exchange rate in a dynamically complete international market wi th two countries and two goods. In his setting finan-cial markets are perfectly integrated and also there is perfect mobility of goods between countries. Dumas (1992) identifies the exchange rate process when there is a transfer cost associated wi th the mobil ity of goods between countries. In similar vein, U p p a l (1993) in a general equilibrium context analyzes the dynamic of international portfolio choice in presence of shipping cost and perfect integrated financial markets. Dumas and Uppa l (2001) and Basak and Croitoru (2003) show how financial markets integration can attenuate the anomalies introduced by imperfection in the good markets. Obstfeld (1994) and Devereux and Saito (1997) study the welfare implications when international risk-sharing are allowed. These cited papers, however relies on the assumption of internationally complete financial markets and/or exogenous real exchange rate. Exceptions to these assumptions are Serrat (2001) and Brandt , Cochrane and Santa-Clara (2002) wi th different objectives from ours. Serrat (2001) studies a two-country exchange economy wi th heterogenous agents and non-traded goods, and shows how the presence of non-traded goods helps explain the home bias puzzle. Brandt , Cochrane and Santa-Clara (2002) examine the relationship between stock market returns and the real exchange rate, and argue that exchange rate is less volatile than the implied marginal ut i l i ty growths from stock market returns. The three papers that are closest to ours are Pavlova and Rigobon (2003), B h a m r a (2003) and Sellin and Werner (1993). Using a model similar to ours, Pavlova and Rigobon (2003) study a range of issues from financial contagion to the effect of demand shocks on stock returns. Our model differs from theirs in that in our model not only the firms in each country can produce several goods with some available in the other country, but also there are investment restrictions. A s a result, the equilibrium asset prices, cost of capital, and the contagion effects we obtain are different from theirs. B h a m r a (2003) also studies the effects of stock market liberalization on equity risk premia, stock return volatilities and the cross-country correlation of stock returns. The cases that he studies, however, are different from ours. Sellin and Werner (1993) consider restrictions similar to ours but focus on international portfolio holdings and the changes in the risk-free rates. Their model don't allow them to analyze the implications of investment restrictions on risky securities prices and agents welfare. Also , there is only one good in the later two models. The two goods feature of our model allows us to study the dynamic of the exchange rate. The remainder of the paper is structured as follows. Section 4.3 describes the economy environment. Section 4.4 gives the results for the bench-market case. Section 4.5 studies the equil ibrium in presence of investment restrictions. We also specialize our results to the special case of logarithmic preferences. Section 4.6 concludes. The proofs are presented in the Appendix. 4.3 The model We consider a continuous-time finite horizon two-country economy. One country is called domestic and the other foreign. They are indexed by d and / , respectively. The uncertainty in this economy is generated by four independent Brownian motions W = (Wi, W2, W3> WAy in a complete probability space ( f i . J 7 , F,V), where F = {ft; t <E [0,T]} is the augmentation of the filtration generated by {W(t) e l 4 ; t <E [0,T]} by the null sets under the probability V. The a-algebra ftt represents the available information set at time t in the economy. A l l stochastic processes to appear in this paper are adapted to F , and a l l equalities involving random variables are understood to hold P-almost surely. We denote by Et the expectation conditional on the information ftt at time t. We define by [x]+ = max(x, 0) the positive part of x. We use ||:r|| to designate the norm of vector x. There are two goods available internationally, wi th processes given as follows d5(t) = 5(t) [ns(t)dt + as(t)TdW(t)] ; and dS*(t) = 8*(t) [fiS.(t)dt + <js*{t)TdW{t)) . We assume that the total amount of good 5 is split between three separate production technologies w i t h dividend processes dSjit) = 5j(t) [pSj(t)dt + aSj{t)TdW(t)] , j = l,2,3. The total consumption good 5 is the sum of these three dividends, 6 = 6i + 52 + £3. There are no frictions in the goods markets. The relative price of the two consumption goods is given by the real exchange rate pit), units of good 5 per unit of good 5*. We w i l l show later that the real exchange rate process follows an Ito process: dp(t) = p(t) [fip(t)dt + ap(t)TdW(t)] . (4.3.1) Each country is populated by a representative agent, whose preference is represented by a time-additive expected ut i l i ty function, where u(c) = c 1 - 7 / ( 1 — 7) and v(c*) = c*^ 1 - 7 ^ / ( l — 7). W h e n 7 = 1 we have the l imit ing case of log util ity. The representative agent in each country is endowed with the good produced in the country. There are three financial securities in the domestic country and two in the foreign country. Domestic country has a risk-free bond and two risky stocks wi th claims to the endowed dividends 6\ and <52. Foreign country has a risk-free bond and a risky stock wi th claims to the endowed dividends 5* and 63. The two bonds are in zero net supply. The total shares of the stocks are normalized to one. The prices of the financial securities are expressed in unit of the good 5. Denote the prices of the domestic and foreign bonds by B, Bf. The prices of the two domestic stocks and the foreign stock are S i , S 2 and Sf, respectively. The market portfolio in domestic country is S<2 = S i + S 2 . The processes of the securities are given as follows: where r(t) and r*(t) are the riskfree interest rates in domestic and foreign countries, respectively. Let dB{t) r(t)B{t)dt, Bf(t) [{r*{t) + nP{t)) dt + ap(t)rdW{t)} , Sj(t) [iij(t)dt + aj{t)TdW{t)] , j = 1,2, Sf(t) [nf(t)dt + af(t)TdW(t)] , dBf(t) dSj(t) + 5j(t)dt dSf{t) + {53(t)+p(t)S*(t))dt li = (r* + fjv,n1,fi2,IJ'f) and E = (ap,a1,a2,af) These are the mean vector and the volatil ity matrix of the risky financial securities. There are five securities (including the riskfree assets) and four sources of uncertainty, therefore the market is complete. We claim that the volatil ity matr ix E is invertible. Since E is endogenous, this property can only be verified in equil ibrium. Unlike in the goods markets, the representative agents of the countries face restric-tions on their shareholdings of the other country securities. We w i l l study two cases. In one case, foreign investors are restricted from holding no more than TT proportion of stock 2 of the domestic economy (i.e., the protected industry stock). In the second case, the agent in the domestic country is restricted from investing no more than n of his wealth in the foreign capital market. 4.4 T h e Bench Market Case - no restriction on assets holding Since the market is complete here, there exists a unique pricing kernel process £ in unit of good 8, d£(t) = - f ( t ) [r(t)dt + 6(t)TdW{t)] w i th f (0) = 1, (4.4.1) where 6(t) is the unique bounded ^-progressively measurable market price of risk. It is given by 0 = E _ 1 ( | U — r l ) , where 1 is the four dimensional vector of 1 when E is of full rank. Here £(t,uj) has the interpretation as the Arrow-Debreu price per unit probability V of one unit of good 5 in state u at time t. Since markets are dynamically complete, the equil ibrium prices of the financial se-curities and the exchange rate in this economy are identical to that in an aggregate representative agent economy where the representative agent has the following aggregate ut i l i ty function: U(8,8*,\) = max [u(cd) + v(c*d)] + A \u(cf) + v(c*f)] , (4.4.2) cd,cf,c*d,c*f subject to Cd + Cf = 8. and cd + c*f = 5*. Here A represents the welfare weight of the foreign agent. The solution of the maxi -mization problem in (4.4.2) is given by cd{t) = 5(t)/(l + A 1 / 7 ) , c*d(t) - 5*(t)/(l + A 1 / 7 ) , Cf(t) = 5 ( £ ) A 1 / T / ( 1 + A 1 / 7 ) ) a n d c){t) = 5*{t)\lli/{\ + A V T ) . The aggregate ut i l i ty function becomes U{5,6*, A ) = (1 + A 1 ^ ) ^ [ ^ / ( l - 7) + <5* ( 1~7 )/ ( l - 7 ) ] . Since the market is complete, A is constant and the allocation is pareto optimal. The equil ibrium in the benchmark economy is characterized by Proposition 4.4.1. Proposition 4.4.1. The equilibrium exchange rate is given by p(t) = 5*{t) The pricing kernel is The equilibrium interest rate and market price of risk are r(t) = /? + 7 M * ) - 7 ( l + 7 ) I M * ) l l 7 2 , 6(t) = 7a 4(t). The riskfree rates in domestic and foreign countries satisfy the so-called uncovered inter-est rate parity relationship r*(t)-r(t) = -pp{t) + 'jas(t)Tap{t). The risk premia are given by lij(t)-r(t) Hf(t)-r(t) 1_ fdSjjt) d5(t)\ 1JtC0V\~s^,l(t)^ Jdt \ Sf(t) ' 5(t) J' 1,2, The representative world investor absolute risk aversion is —U$s/Us = ~y/5 and its absolute prudence coefficient is —Usss/Uss = ( l + 7)/<5. The riskfree rates are decreasing wi th the consumption growth volatilities and the precautionary savings of the agents. The market prices of risk are proportional to the consumption growth volatilities vector. The intuit ion is as follows. W h e n the aggregate consumption is more volatile, agents demand more safe security for hedging, which induces them to lower their risky stocks holdings. In equil ibrium, the interest rate decreases, creating an extra incentive for agents to hold risky securities. The risk premia on the other hand, equal the aggregate absolute risk aversion times the covariance between stock returns and the consumption growth, which is the standard International C A P M obtained wi th complete market assumption. The uncovered interest rate parity establishes the relationship between the domestic and foreign riskfree rates, and the exchange rate. Indeed, consider two investments, one in domestic bond and the other in foreign bond. The return on investment in foreign bond is equal to the return on investment in domestic bond minus the drift of the exchange rate plus an additional risk premium. The additional risk premium in the expression of the interest rate differential appears only in the dynamic setting. It is the compensation for exchange rate risk. The pricing kernel in unit of good 5* can be obtained using the exchange rate as follows £*(£) = p(i)£(t), which is the law of one price relationship. This relationship is a no-arbitrage condition. The price of domestic stocks are Sj(t) = Et i: £(£) 5(t) 5j(s)ds J = 1,2. W h e n Lis, and as are non stochastic, Sj is simplified as follows: SM-W) £eKp{J\-(3+»8I(T)-w^ The domestic aggregate equity is Sd(t) — Si(t) + S ^ t ) . The price of foreign stock is Sf(t) = Et e'"^ ) (p(sW(s) + 53(s))ds W h e n the aggregate consumption goods processes follow geometric Brownian motions, , 1 - exp { - ( / ? - ( ! - 7 K + 7(1 - 7 ) I N I 7 2 ) (T - t)} sd(t) = 8(ty -Ek Sf(t) = p{t)5*{t) / ? - ( l - 7 ) M 5 + 7 ( l - 7 ) l k 5 | | 2 / 2 * ( * ) N _ 7 53(s)ds 1 - exp { - ( / ? - ( ! - 7 W + 7(1 - 7) W l l 2 / 2 ) (T - t)} (3 - (1 - 7 K + 7(1 - 7 ) l l 2 / 2 « j ( s ) V 7 If each country specializes in one good, i.e., domestic firms produce only good 5 and foreign firms produce only good 5*, then for logarithmic preference agents, countries' equities are Sd(t) = Sf(t) = (1 — e~^T~^)S(t)/p. This has been shown by Cass and Pavlova (2003), Pavlova and Rigobon (2003) and Zapatero (1995). This perfect conver-gence between the two markets occurs through the exchange rate representing the terms of trade between the two countries. The intuit ion is as follows. Suppose a positive out-put shock on the domestic economy. It causes the return on domestic equity to increase, but at the same time deteriorates the terms of trade for the domestic country. W h i c h means the terms of trade is improved for foreign country. Foreign output value increases, thereby boosting the foreign equity value. To avoid this redundancy we have to assume that foreign equity gives c laim to the two goods, thus the market is complete. In our setting, even under the assumption of log preferences, the correlation between the two equity markets is less than one. To see that, without loss of generality lets assume /J,$3/S = 0, thus under log preference assumption Sd(t) — 1 - e ^T *] (5(t) — 53(t)) and Sf(t) = i-e 0 ( T t] (fify + S3(t)). Hence the equity volatilities are 00-5 - S3crs3 Sio~$l + 52crS2 5a$ + 53os3 a d = 5-53 = 8i + S2 a n d a f = 5 + 53 The correlation between the two equity markets' returns is: * l k « l l a - g f l k » l l a Sias1 + 52cr^2 + 2<53i7,s3 5i + 82 + 2<53 P = < 1. k d l l l k / l l lk°"<5 + ^ 3 l l l k ° r < 5 - h°63\\ From Schwarz inequality, this correlation is less than 1 and can be negative. For example, suppose that aJaSs = 0, then p = (82\\a5\\2 - ^ I k & l H / O P l K H 2 + ^ | K | | 2 ) < 1. 4.5 Equi l ibrium with Investment Restrictions A s we mentioned earlier, two cases of investment restrictions are considered. In the first case, foreign investors are capped in their shareholdings of the protected sector of the domestic economy. We assume stock 2 to be the stock of the protected industry in the domestic economy. Thus, foreign investors cannot hold more than TT proportion of stock 2. This case is analyzed in section 4.5.2. In the second case, domestic agents are restricted in their foreign investments. They cannot invest more than TT of their wealth in foreign capital (equity and bond). We explore this case in section 4.5.3. 4.5.1 Investors' optimization problem The wealth process of agent i (i = d, / ) is where 7Tj = (7r,0, ^ i , 7ri2,71"^) is the vector of percentage investments in foreign bond, domestic stocks 1, 2, and foreign stock, respectively. We denote by ^ the pricing kernel facing the domestic agent and by the one facing the foreign agent in the fictitious complete markets described in the Appendix . A s Cox and Huang (1989), agent i maximization problem can be restated as the following variational static problem: A t optimum, the marginal rate of substitution on one unit of consumption good is pro-portional to the fictitious unit pricing kernel: e~^u'(ci(t)) = yiCi(t) and e~^V(c*(£)) = yip{t)ii(t). Hence, the optimal consumptions are: Cj(£) = [e^y^^t)] ^ 7 and c*(t) = [e^yipfy^t)] w i th yi satisfying the budget constraint. dXi(t) = Xi(t) [r{t) + 7n(t)T(fl(t) - r(t))] dt -[a(t)+p(t)c*(t)}dt + Xi(t) [vr i(t)TS(t)] dW(t), under the budget constraint: Definition 4.5.1. A n equil ibrium is a collection of optimal {7Tj(.), CJ(.), C * ( . ) , i G {/, d}} and {r(.), /x p(.), >u.i(.), /z 2(.), a p ( . ) , oi( . ) , a 2 ( . ) , cr/(.)} such that, 7r, finances c*(£), £ € [0,T]}, and the goods and securities markets clear V i G [0,T]: cd(t) + cf{t) = S(t), ( G o o d £ ) cW) + c»f(t) = 5*(t), (Good 5*) TTdi(t)Xd(t) + 7rfl(t)Xf(t) = Si(t), (Domestic Stock 1) vr d 2(£)X d(£) + irf2{t)Xf(t) = S2{t), (Domestic Stock 2) ndf(t)Xd{t) + TTff{t)Xf(t) = Sf(t), (Foreign Stock) Vdo(t)Xd(t) + irf0{t)Xf(t) = 0, (Foreign Bond) (1 - lTTrd(t))Xd(t) + (1 - lTirf(t))Xf(t) = 0. (Domestic Bond) 4.5.2 Restriction on foreign investors The domestic government imposes investment restrictions on foreign investors in the domestic market to protect particular industries i n the domestic economy. We assume stock 2 to be the stock of the protected industry in the domestic economy. Hence, foreign investors are restricted from holding more than n proportion of the domestic stock 2. The representative investor is constructed similar to Equation (4.4.2). However, be-cause of the constraint imposed on foreign investors, markets are not complete anymore. Therefore, the weighting parameter A assigned to foreign agent by the representative i n -vestor becomes a progressively measurable process wi th respect to the filtration F . This weaker notion of aggregation has been first introduced by Cuoco and He (1994) in the context of incomplete markets, and been used by Basak and Cuoco (1998), Basak and Croi toru (2000), and Shapiro (2002) among others. Assuming the equil ibrium exists, the proposition below characterizes the general equilibrium. Proposition 4.5.2. The weighting process A is characterized by d\{t) = A(t) [fix{t)dt + ax{t)TdW{t)] , (4.5.1) withiix(t) = ja5{t)Tax{t) + \\ax(t)\\2/{l+X{ty^)+n{t)x2{t) andax(t) = -x2(t)i:-\t)L3, where i3 = (0, 0,1, 0 ) T and x2{t) > 0, V£ G [0,T]. The pricing kernel is The exchange rate is m = e \i + x(oyh) [W)J ' ( } TTte interest rate, r, and market price of risk, 9, are: r(() = /3 + 7 « ( ! ) - 7 ( l + 7 ) I M i ) l l 7 2 - ^ ( T 7 ^ ? l | E " 1 ( t ) 1 3 l | ! f e ( 1 ) ) 2 - T T m ^ ^ ^ \(tVh 9{t) = 1as(t)+ 1 + y s a ^ S - ^ K (4.5.5) T/ie interest rate differential is r*(t)-r(t) = -Lip(t) + <ya5(t)Tap(t). (4.5.6) TVie rofc premia are « W - r ( t ) = ^ ^ . - j i i j , (4.5.7) It is worth noticing that our exchange rate process in Equat ion (4.5.3) is independent of the weight A. Therefore, the exchange rate is explicitly characterized wi th the two aggregate output processes parameters. 3 Even under this setting of incomplete market, the uncovered interest rate parity formulation in Equation (4.5.6) doesn't change. The risk-free rate, market price of risk and risk premia are al l function of the variable x2. It is positive when the constraint binds, and equal to zero otherwise. We solve for 3 T h i s is in part do to our utility function specification, however for more complex utility specifications for the agents, the exchange rate can be function of A. the explicit expression of x2 in the case where agents have logarithmic preferences. The results for this special case are given in Corollary 4.5.5 below. Proposit ion 4.5.2 identifies several important differences between the bench-market and the current economy where foreign investors are restricted in their holding of the stock of the protected industry in the domestic country. F i rs t , foreign investors because they cannot hold more of the protected stock demand more bonds. The overall demand for bonds increases, then bond prices go up and drives down the risk-free rates. Since X(t) is t ime-varying because of the incompleteness of the market, the, risk-free rate is more volatile than that in the benchmark economy. For example, if the aggregate dividend 8 follows geometric Brownian process, the risk-free rate in the benchmark economy is constant, while the risk-free rate in the restricted economy is time-varying. Second, the market price of risk of the protected industry is increasing, while that of the unprotected industry is ambiguously linked to its correlation degree wi th the protected sector. Because of the restriction imposed on foreign investors, domestic agents have to hold the remaining supply of the protected industry's stock. Thus, the market price of risk of the protected stock goes up to compensate domestic investors for the excess holding. T h i r d , the risk premia deviate from the standard i n t e r n a t i o n a l - C A P M obtained in the bench-market. The cost of capital of the protected sector in the domestic economy is higher than that of the unprotected sector al l else equal. 4 Stock prices and volatilities are characterized by the following lemma. L e m m a 4.5.3. The price of stock j is 4 I f there are financial derivatives in the market, institutional investors wil l use them to overcome the restrictions on stock holdings imposed on them. By taking positions on the derivatives markets they can attenuate substantially the effect of the investment restrictions. The existence of financial innovations can be addressed within this framework. For that purpose, stock 2 wil l be the real asset and the risky security 1 the financial derivative. Depending on the level of correlation between the two securities the constraint wil l have less or more impact. If the two securities are highly correlated then the constraint effect is negligible since agents wil l use security 1 to overcome the holdings restrictions on stock 2. This practice is current in international financial markets, examples are the cloned or mirror funds in Canada. where £ is the pricing kernel. We denote by DtY(s) the Malliavin derivative of the variable Y(s) at time t.5 Assuming all the regularity conditions are satisfied for the Malliavin derivatives to exist, the volatility of stock j is given as follows: jtTas)vMs)ds Et It i{s)5j{s)ds + Et '^8j(s)(das)/d8)Bt8(s)ds Et It'Z(s)8j(s)ds +-Et '^8j(s)(das)/dX)-DtX(s)ds Et It Z(s)6j(s)ds Country 's stock price is the expected discount value of its future dividends using the pricing kernel. The stock returns volati l ity has four terms. The first term is the volati l ity of the pricing kernel. The second term is the impact from the perturbation of the dividend process of the stock. The th i rd and fourth terms show how shocks on the aggregate dividend and the weight affect the stock volati l ity through the pricing kernel perturbation. One main difference between the volati l ity of the stocks in this restricted economy and the bench-market is the last term. Indeed, in the bench-market, the last term is equal to zero since A is constant. Below we provide approximate closed-form expressions for the stocks volatilities when agents have logarithmic preference. Agents are facing different investment opportunity sets. Thus, their consumption streams are different. The following corollary characterizes the consumption process of the agents. C o r o l l a r y 4.5.4. The processes of agents' consumptions of good 8 are dd(t) = a(t) [nctWdt + aCi(t)TdW{t)) , i = d,f, 5 We characterize the stock volatility using the Malliavin derivative chain rule. The Malliavin deriva-tive of a variable X denoted by DtX measures how perturbing is X to an innovation in the Brownian motion vector at time t. The Malliavin calculus has recently been applied in finance by Ocone and Karatzas (1991), Detemple and Zapatero (1991), Serrat (2001) and Detemple, Garcia and Rindisbacher (2003) among others. We refer the interested reader to the monograph by Nualart (1995) and Oksendal (1997) for the existence conditions. •with \{t)lhTt(t)x2{t) Vet® = ns(t) -7 ( l + A(t)V7) ( 1 + ^ 1 / 7 W ^ ) + ( 1 " A ( t ) i y 7 ) l k A ( t ) " a 7(1 + A(t)V7) V o w A V / 2 7 ( l + A(t)V7) - + 7 ( 1 + A ( i ) 1 / 7 ) , (1 + 7) / m x m , ( i - A ^ I M O H 2 + l~i i U i \ l / ^ O M * ) 7 ( l + A(t)V7) V ° ^ 2 7 ( l + A(i)V7) anci A f t ) 1 / 7 a c / ( t ) = ^ ( O - ^ ^ ^ ^ x a f f l S - 1 ^ -Similarly, the processes of their consumptions of good 5* are dc*(t) = c*(t) [^(t)dt + a c .(t) TdW(t)] , i = dj, with fxc* and ac* obtained by replacing 5 by 5* in the expressions of \iCi and aCi, respec-tively. Agents' consumption rates volatilities are proportional to the volatilities of the fic-titious pricing kernel they are facing. Because of the constraint imposed on foreign investors, the two representative agents face different investment opportunity sets, there-fore, their marginal rates of substitution are different. They pick differently their con-sumptions. In this economy, agents' consumption growths are not perfectly correlated with the aggregate consumption growth as it is in the benchmark economy. Shocks on the protected stock process in the domestic economy tend to have asymmetric effects on the consumptions of the agents, domestic agent who hold more of that stock is more affected than foreign agent. Log utility case In order to better understand the economic implications and also provide approximate closed-form solutions for stock volatilities, we consider the special case of time-additive logarithmic preferences. Because of the log uti l i ty feature, the total wealth of the world is S(t) = 5 i ( t ) + S2(t) + Sf(t) = 2(1 - e-W-^Mtyp. Corollaries 4.5.5 and 4.5.6 characterize the equilibrium when agents have logarithmic preference. C o r o l l a r y 4 .5 .5 . The weighting process becomes: _ A(t ) ( l + X(t)) S2(t) [A(t)/(1 + \(t)) - 7f (t ) ] +  d X { t ) ~ - | | £ - ^ 3 I I 2 S(t) A(t) / (1 + X(t)) { t ) ) d W { t ) ' where A(0) = Xf(0)/Xd{0) and t3 = ( 0 , 0 , 1 , 0 ) T . The equilibrium interest rate, r, and market price of risk, 9, are r(t\ fi+„(t\ l U m i P M t ) (S2(t)\2[X(t)/(l + X(t))-7T(t)] + r(t) = P + ,s(t)-hs(t)\\ - | | E _ 1 ( t ) t > | | a ^ j A(t) / (1 + A(t)) ' • ,9m T „ , A(t) S 2 ( t ) [A(t)/(1 + X(t)) - 7 f ( t ) ] + ^ _ 1 m = a s { t ) + W T W ^ W — A ( O / ( I + A ( O ) — E ( T ) T 3 ' T/ie nsfc premia are: Hi(t)-r{t) = jtcov 1 fdS2(t) d5(t)\ X(t) S2(t)[X(t)/(l + X(t))-n(t)Y Si(t) ' 8(t) dS2{t) (t) m' S(t) dSf(t) d5(t) sf(t) 5(t) dt V S2(t) ' 8(t) ) ||S-i(t) i 3|| 2 S(t) A( i ) / (1 + X(t)) In the expression [A(i)/(1 + A(i)) — 7 f t h e term A( i ) / (1 + X(t)) represents the proportion of stock 2 foreign investors would hold if there were no holding restrictions. The volati l ity matrix is determined below using perturbation methods. There is a poor risk sharing wi th respect to the protected industry stock. Indeed, because of the restriction imposed on foreign investors, on one hand domestic investors hold excess shares of the protected sector. Thus the market price of risk is adjusted to compensate domestic investors for this excess holding. O n the other hand, foreign investors invest more in bonds, which reduces the volati l ity of their consumption (see Corollary 4.5.4 for agents' consumption volati l ity) . The risk-free rate, market price of risk and risk premia variations are function of the relative size of foreign investors' wealth (or foreign country size) measured by A. The impact of the restriction is proportional to the relative size of foreign wealth. Corollary 4.5.6 below provides approximate closed form solutions for stock volatilities. In general, even in this simplified case of log preferences, we cannot solve and obtain explicit closed-form solutions for stock volatilities. Basak and Gallmeyer (2002) derived the Forward Backward Stochastic Differential Equation ( F B S D E ) for the stock price and use numerical methods to obtain their results. We instead use perturbation methods [see J u d d (1998)] to obtain approximate closed-form solutions. To achieve that, we first derive the F B S D E of the price processes. Second, we use continuation methods to solve for stock prices. T h i r d , we use the expression of stock prices and solve the simultaneous equations of volatilities. The different steps of this procedure are exposed in details in the Appendix. For a discussion on the precision of the perturbation methods, we refer the interested reader to Zeidler (1986). 6 Lets denote by the superscript "o" quantities from the benchmark economy. For example, S? is the price of the domestic stock j in the benchmark economy, and CT? its returns' volatility. W h e n fj,g = ^ — fig — aj(asj — as) is non stochastic, S°(t) = 8j(t) J * t T exp{f*(—j3 + psj/s{T))d'T}ds. To derive the stock prices and volatilities, we assume that ^ = p§3/s = 0, then S°2{t) = ^j^-82(t), S°d(t) = i - e ^ " (8(t)-S3(t)) and S°f{t) = 1 ~ e " ^ ( r " t ) (8(t) + 83(t)). We also use e = [ A / ( l + A) - T T ] + / { A / ( l + A)}. 6 T h e expression of the variable y is written as the Taylor series expansion of 6 as follows: y = Vo + E£=i + 0(eN+1), where 0(eN+1) means that \\y - [y0 + E^=i I/n£n]||/lkir+1 < oo. We use the F B S D E satisfied by the price process to solve for the coefficients of the power terms. Once we obtain the parameters estimates, using stochastic calculus, we write the stocks volatilities as a system of simultaneous equations, which we solve. C o r o l l a r y 4.5.6. Stock prices are 5 r ( t ) com , Mt)S{t) {S°2(t)\ bl{t)+\\z°-i(tU\2\s(t)J 2(1 S(t) ) ( l - l o g ( ^ ) ) ( l + * $ ) ( l - l o g ( * $ ) ) Ws3/s(tW e + 0(e2), S2(t) = S2°(t) 1 -2A(t) K i - f ^ ) f ? d - i o g ( ^ ) ) e 5(t) ' S ( i ) 6 * w " b d [ t ) \wS3/m2\\^-m^\\2\s(t)) i + Sf(t) = S°f(t) + \(t)S(t) ||aw,(i)||2||S°-i(t).3|p + 0(e2), 5(t)J l - l o K f § @ ) ) c + 0 ( ^ 1 - lOE ' 6(t) e + 0(e2 Stock volatilities are 'I \S Sl\\Y,o-h^\S + 2 ( l - f ) ( l - l o g ( f ) ) ( l + f ) ( l - l o g ( f ) ) 1 I K 2 / < s l l 2 II °S3/61|2 + r 2 ( l - f )(1 - 21og(f) ) - 2 f (1 - l og ( f ) ) l l ^ l l 2 ( l + f ) ( l - l o g ( f ) ) II^AII2 2A (aS2 - a5)e \ + 0(e2), 0-2 o-d + A S fS°2 \\cTS3/s\m°-h3\\2 s°d \s A 1 , <*31 l + T l o g ( T ) K - a ( 5 ) e + 0 ( e2 ) , a) + S fS°2 \\aS3/s\m°-%\\2 S°f \S A 5 fS°2 l + 54 \ (l-\og{^)){2a&2-a°f-a5)e l + y l o g ( T ) (aSs - as)e + C ( e 2 \\as3/s\m°-h3\\2 S°f \ S The price of the domestic protected stock decreases, which affects the domestic equity value. The changes on stock volatilities are ambiguous. In the current economy, because of the investment restriction, the two market equities volatilities move in opposite direc-tions. The domestic sectors are also affected differently. Thus by protecting one sector of the domestic economy, policy makers are able to reduce the exposure of other sectors to shocks in the international capital market. Pavlova and Rigobon (2003) develop a model in which policy makers by taking decisions affecting directly demand shocks can influence the contagion effects between countries equity markets. In this framework, i n -stead, policy makers by imposing artificial investment restrictions on agents control the financial contagion effects. To analyze that more closely, lets compute the correlation between domestic and foreign equity using the following formula: P = °d<rf/\Wd\\\Wf\\-We can rewrite the stocks' volatilities as follows: Sdad = S°da°d -Ae + 0(e2), Sfaf = S°fa°f + Ae + 0(e2), with (S3, ,h-o-§3/s\\2\\X°-h3\\* \ SJ 1 V SJ V "5 l + l - l o g ( - ^ ) \ ( 2 a s 2 - a s Using this volatil ity expressions, the correlation is given as follows: ( 1 P° \ e + 0{e2 Table 4.1 presents simulated correlations between the domestic equity market returns and the foreign one. Depending on the in i t ia l parameters values, the correlation between the two market returns can either be higher or lower than the benchmark level. For example, in panel (a), p < p°, and in panel (b), p > p°. Thus by imposing restrictions of foreign investors in some sectors of the domestic economy, countries can reduce the contagion effect on their economy measured by the correlation between markets' returns, p. Table 4.1: Simulated correlations between domestic and foreign equity returns. This Table shows the simulated correlations between the domestic and foreign market re-turns. The values of the parameters used in the simulations are: T = 6 years, ps-t = us2 = uS3 = us- = 0.05, <5i(0) = 200, S2(0) = 53{0) = 5*(0) = 100. In Panel (a), aSl = (0.15,0, -0.05, - 0 . 10 ) T , a&2 = (-0.05,0.15,0, - 0 . 05 ) T , a&3 = (-0.05,0.05,0.15, - 0 . 1 0 ) T , as* = ( -0 .05, -0.05, -0.05,0.15) T . In Panel (b), a5l as3 = (0,0,0.15,0) T , as* = (0,0,0,0.15) T . (0.15,0,0,0) T , a52 = (0,0.15,0,0) T , Values of p° Panel (a): Values of p T-t 0.10 7T 0.30 0.50 T- t 0.10 7T 0.30 0.50 5 4 3 2 1 0.7373 0.7341 0.7348 0.7353 0.7359 0.7374 0.7346 0.7334 0.7306 0.7307 0.7389 0.7398 0.7403 0.7369 0.7360 5 4 3 2 1 0.7067 0.7066 0.7103 0.7133 0.7158 0.7185 0.7178 0.7185 0.7175 0.7192 0.7316 0.7333 0.7346 0.7317 0.7314 Values of p° Panel (b): Values of p T- t 0.10 7T 0.30 0.50 T - t 0.10 7T 0.30 0.50 5 4 3 2 1 0.7428 0.7402 0.7409 0.7406 0.7410 0.7428 0.7431 0.7440 0.7432 0.7421 0.7444 0.7463 0.7451 0.7433 0.7441 5 4 3 2 1 0.8724 0.8691 0.8684 0.8686 0.8688 0.8235 0.8237 0.8248 0.8252 0.8250 0.7752 0.7767 0.7757 0.7740 0.7753 Welfare analysis Assuming agents have logarithmic preferences, the welfare of agent i can be decom-posed into two components as follows: £/, = U° + AUi, i € {d,f}, where U° is the welfare level in the bench-market and is given as follows U° (1 - e "^ ) [21og((l - e^T)li3Xm) + log(«J(0)/<J*(0))] //? +E [ e-* [ {ps(T) + ps'(r)-(\\as(T)\\2 + \\as*(T)\\2)/2)drdt Jo Jo Uo Jo and AUi is given in the following corollary: C o r o l l a r y 4.5.7. Agents' welfare variations AC/j are 2 ( l - e - ^ ) l o g ( X d ( 0 ) / X d ° ( 0 ) ) g a ( r ) [A(r)/(1 + A(r)) - 7f(r)] + S(r) ||E- i (r ) t 8 ||A(r) / ( l + A(T)) droit AU, 2 ( l - e - ^ ) l o g ( X / ( 0 ) / X / ° ( 0 ) ) -E f A l + 2A(r)) 5 2 ( r ) [A(r)/(1 + A(r)) - T T ( T ) ] + 5 ( r ) ||E-i (r ) t 3 ||A(r) / ( l + A(r)) } 2 Jo Jo The welfare variation expression in Corollary 4.5.7 has two components. The first component captures the change in the in i t ia l wealth by imposing the investment con-straint on foreign investors, and the second component represents the hedging gain/loss relative to the benchmark holdings. Recal l , in the economy setting, we assume each agent to be endowed wi th the in i t ia l supply of his home country. From Corollary 4.5.6 above, since domestic stock value decreases and foreign stock value increases, the in i t ia l wealths A^(0) decreases and -AT/(0) increases from their benchmark levels. However, the drift of the changes in the relative proportion of foreign agent wealth in the economy is negative: Et [ d {A ( i ) / ( l + A(t))}] = Et [d {Xf(t)/(Xd(t) + Xf(£))}] < 0. Thus foreign agent wealth is decreasing, which implies that the negative wealth effect in domestic agent welfare variation disappears over time. There is a welfare shift from foreign agent to domestic agent. Indeed, the protected industry idiosyncratic risk is shared poorly. It benefits domestic investors who gain higher returns from the protected industry stock. The intuition is as follows. Suppose that domestic and foreign agents have equal in i t ia l wealth. Assuming domestic agent has a l l the domestic endowments. In the benchmark economy, both agents w i l l hold half of the stocks. There is no lending or borrowing. In the restricted economy, since foreign investors are restricted from holding more of the domestic protected stock, the demand for bonds increases and there is less international demand for domestic stock. Therefore the price of domestic stock decreases and the risk-free rate decreases as well. Thus domestic investors can borrow cheap, and they w i l l do so to satisfy their foreign stock holdings. There are two things happening here: domestic investors borrow cheap to invest, therefore they increase their opportunity sets; and secondly since the price of domestic stock decreases, domestic investors keep their holdings and benefit from the expected high future returns. In sum, protecting a particular industry creates at least two opposite effects a l l else being equal. The first (negative) effect is the increase of the cost of financing for the protected sector. The second (positive) effect is the increase of domestic agents' welfare. 4.5.3 Restriction on domestic residents In this economy, we assume that domestic residents cannot invest more than TT of their wealth in foreign capital market. In the previous economy, we have constructed the representative investor by assigning a stochastic weight to the constrained foreign investor. Here, instead, the stochastic weight is assigned to the constrained domestic investor. Therefore, the representative agent uti l i ty is constructed as: Cd + Cf = S and c*d + c*f = 5*. The optimal consumptions are cd = 5X^/(1 + A 1 / 7 ) , c*d = 5*X1^/{1 + A 1 / 7 ) , cf = 5/ (1 + A 1 / 7 ) , tif = 5*/(l + A 1 / 7 ) . Substituting back cd, Cf, c*d and c*f in the ut i l i ty function expression yields: U(6, 5*, X) = ( l + A 1 / 7 ) 7 [ ^ / ( l - 7) + / ( l - 7)]. Remark 4.5.8. Even though the representative investor utility formulation is similar to the one obtained in the previous economy, the weight X is not the same. Assuming the equil ibrium exists, its characterization is given below. Proposition 4.5.9. The weighting process X is characterized by U{5, 6*, A) = max ^ [u{cf) + v(c*f)] + A [u(cd) + v(c*d)}, (4.5.10) subject to dX(t) = X(t) [nx(t)dt + ax(t)TdW(t)] (4.5.11) withfix(t) = 7 ^ ( i ) T a A ( t ) + ||a A(i)|| 2 /(l+A(i) 1 / 7)+7f(i)a;W andax(t) = - x ( t ) E " 1 ( t ) ( 4 i + t 4 ) , w/iere t l = ( 1 , 0 , 0 , 0 ) T ; t 4 = ( 0 , 0 , 0 , 1 ) T ond x(t) > 0, Vt € [0,T]. The pricing kernel is T/ie exchange rate is ITie interest rate, r, and market price of risk, 6, are: r(t) = /3 + 7 M , ( t ) - 7 ( l + 7 ) | | a , ( i ) H 7 2 - 1 + A(i )V7 w v ; _ ( l + 7 ) A i * ) ^ ] _ | | s - 1 ( t ) ( t l + i 4)|| 2(x(t)) 2, (4.5.14) 27 (1 + A ( i ) V 7 ) 2 " v 4 ; " V W ; V ; A ( W / 7 = ias(t) + + y ^ S - 1 ^ ) ^ + . 4 ) . (4.5.15) TTie interest rate differential is X(t)1^ r*(t)-r(t) = -P,p(t) + 1as(t)Tap(t) + T ^ - ^ x ( t ) . (4.5.16) The risk premia are 1 ,dSi(t) d6(t). r(t) = ^ c o ^ , ^ ) , (4.5.17) / i 2 ( 0 - r ( t ) = (4.5.18) 7 / N 1 ,dSf(t) dS(t). A(t) 1 / ' Similar to the previous economy, the riskfree rates, market price of risk and risk premia are al l function of x. The variable x is positive when the constraint binds, and equal to zero otherwise. We provide the explicit expression for x in Corol lary 4.5.11 below when agents have logarithmic preference. Proposit ion 4.5.9 identifies several important differences between the bench-market and the current economy where domestic residents are restricted in their investment in foreign capital market. F irs t , domestic agents restricted to hold less foreign capital demand more domestic bonds, hence the overall demand for domestic bond increases, which drives its price up, and then the riskfree rate is lowered. O n the other hand, the difference between the foreign riskfree rate and the domestic one is augmented by the factor x A 1 / 7 / ' ( I + A 1 / 7 ) . Therefore, the uncovered interest rate parity relationship doesn't hold anymore. Aga in X(t) is time-varying because of the incompleteness of the market, thus the riskfree rate is more volatile. For example, if the aggregate dividend 5 follows geometric Brownian process, the riskfree rate in the benchmark economy is constant, while in the restricted economy it is time-varying. Second, the market price of risk of foreign equity is increasing, while the market price of risk of domestic stocks are adjusting by their correlation degree wi th foreign stock and the exchange rate. Because of the restriction imposed on domestic agents, foreign agents have to hold the remaining supply of foreign equity. Thus the market price of risk of foreign equity goes up to compensate foreign investors for the excess holding. T h i r d , the risk premia deviate from the standard i n t e r n a t i o n a l - C A P M obtained in the bench-market. Indeed, by restricting its residents' investments abroad, the domestic country is able to keep capital inside. The domestic market cost of capital is lowered, al l else being equal. Since domestic investors are capped in their foreign investment, the two representative agents face different investment opportunity sets. Thus, their consumption streams are different. Corollary 4.5.10. The processes of agents' consumptions of good 5 are da(t) = a(t) [ncifidt + aCi{t)TdW{t)] , i = d,f, with 7 ( 1 + A ( £ ) V 7 ) + ^ 1 t J ? ^ U o M * ) + ( 1 " A ( t ) V 7 ) I M t ) l | a 7 ( l + A(t)V7) V ° v 7 A W 2 7 (1 + A(t)V7) " " 7(1 + A(t)V7) (1 + 7 )A(t)V7 / ( 1 _ A(t)^)|k A (t)|| 2 and 7 ( l + A(t)V7) y°^> 2 7 (1 + A ( i ) 1 / 7 ) ffca(*) = ~ a ^ ) 7 ( 1 + ^ ( t ) i / 7 ) S 1 ( i ) ( t i + t4), crc/(t) = a,(t) + x(t) + V j[ S - 1 ^ ) ^ ! + .4). Similarly, the processes of their consumptions of good 5* are dc*(t) = c*(t) [pc*(t)dt + ac:(t)TdW(t)] , i = d,f, with fjLc* and o~c* obtained by replacing 5 by 5* in the expressions of \iCi and aCi, respec-tively. Agents' consumption rates volatilities are proportional to the volatilities of the ficti-tious pricing kernel they are facing. Because of the cap imposed on the domestic repre-sentative investor, his marginal rate of substitution is different from the foreign investor's one. Domestic and foreign agents pick differently their consumptions. The consumption growths are not perfectly correlated wi th the aggregate consumption growth as it is in the benchmark economy. Shocks on foreign stock market w i l l affect more foreign agent's wealth than domestic agent's wealth. L o g u t i l i t y case Aga in to better understand the economic implications and provide approximate closed-form solutions for stock prices and volatilities, we consider the special case where agents have time-additive logarithmic preferences. Since we are in a log preference setting, the total wealth available in the world is S(t) = S^t) + S2(t) + Sf(t) = 2(1 - e-KT-*)6(t)/l3. Corollaries 4.5.11 and 4.5.12 characterize the equil ibrium in the logarithmic preference case. C o r o l l a r y 4 .5 .11 . The weighting process becomes dx(t) = - i i s - ^ ^ J i p [Sf(t)/s(t) - 7 f ] + ( t l + t 4 ) T ( s - 1 ( t ) ) T d i y ( t ) , where A(0) = Xd(0)/Xf(0), n = ( 1 , 0 , 0 , 0 ) T , and tA = ( 0 , 0 , 0 , 1 ) T . The equilibrium interest rate, r, market price of risk, 9, are «{ ( ) = „ , ( t ) + A(t) [ t i - / f f ( ! ~ S E " ' " ) ( t l + ^ y i . x ( t i + i.4j|r T/te market equity risk premia are: 1 fdSJf) dS(t)\ Mt)-r(t) = - c ^ - ^ . - ^ - j , Mt)-r(t) = _ j + A ( t ) n s _ 1 ( t i + t 4 ) | | 2 • From the interest rate differential equation in Proposition 4.5.9 and the corollary above, the foreign country risk-free rate is r' = f> + « • - I M i 2 + A (1 - S,/S) Thus the risk-free rate in the foreign country is higher than its benchmark level. From the pricing kernel volatilities, foreign investors face a higher exposure to foreign equity idiosyncratic risk than domestic investors. The explanation is that, since domestic investors are restricted to hold less of foreign equity, foreign investors hold the excess supply of their market equity while domestic investors increase their holding in bonds. The market price of risk is adjusted to compensate foreign investors for their excess holding. Domestic investors demand more domestic bonds for precautionary savings. It drives the domestic risk-free rate and their consumption volati l ity down (see Corollary 4.5.10 for the expression of agents' consumption volati l ity) . The risk premium on foreign stock increases and that on domestic equity decreases, lowering the cost of capital in the domestic market. There is a perfect risk sharing of domestic equity, but not of foreign equity. The cost of capital in the domestic market becomes lower, since more domestic residents are keeping their capital home. The re-striction placed on domestic investors has impact on excess stock returns proportional to the relative size of domestic investors (or domestic country) measured by A. Corollary 4.5.12 below provides approximate closed-form expressions for stock prices and volatilities. The steps to follow in order to obtain these expressions are similar to the previous case and can be seen in details in the Appendix . Lets denote by the superscript "o" quantities from the benchmark economy. For example, S° is the price of the domestic stock j in the benchmark economy, and cr? its returns' volatility. To compute the stock prices and volatilities, we assume ps^/s = Ps3/5 = 0. We also use e = [Sj/S — 7f] + . C o r o l l a r y 4 .5 .12. Stock prices are X(t)Sj(t) S^t) = S1°(t) + s2(t) sd(t) sf(t) (1 * G l ) ( l - l o g ( * $ ) ) S(t) . E ' - i ( t ) o 1 + t 4 ) i i n \Ws3/S(tw 2 a ) ( l - l o g ( % g ) ) l s(t) Mm2 r + 0 { e h S°(t) 2A(t) ( l - log(Sffi)) s°2(t) s°d(t) s°f(t) 1 + 1 + SQL) \\aS2/5(t)\m°-i(t)(L1 + cAW m \\aS3/s(t)\mo-x{t){tl+LiW + 0{e\ ^ ) ( l - M * > ) ( l - l o g lk,3 / 4(t)|| 2||E»-i(t)( t l + t 4 ) - M | g ) + 0(e2 Stock volatilities are XS°f ( l - f ) ( l - l o g ( f ) ) c ° ( l - l o g ( f ) ) 5f||E«»-i(4l + t4)||2 F<53/<5| - 2-S \as2/s\ (a°f-aSl)e \S°f 51°||E°-i(ii + i 4 ) +2 AS? l - | l o g ( | ) . k & / * . l 0-2 Sf||S°-i(ti + t4)||2 5 ||a, 2 / c J|| 2 ' 2 A ( l - l o g 4 ) ) ( ^ - a , ) £ 2 ' 5 l l a ^ P H E - i ^ i + ^ l l 2 2A S°f S ||a52/5||2||E°-i(^i + t4)||2 A 0 « a - a6)e + G(e2), S°f / i53\ / ,(5; rf ' V , 3 / , | | 2 | | S - i ( i l + t4)||25d 0 A S°f k a s / ( 5||2||E»-i( t l + t4)||25S A kov* ! K 1 - 1 ) I 1 - > < > ) < • ? {a6s -a6)e + 0{e2), (aS3-as)e + 0(e2). 1 ^ 3 1 l - T l o g ( y ) 1 ^ 3 1 ~ 7 s ^ Our results reveal that foreign equity value decreases and the domestic equity value increases as the result of the equilibrium. The intuit ion is as follows. Because of the restriction imposed on domestic residents, they increase their equil ibrium holding in domestic equity, which pushes its price up. However, the impacts on stock volatilities are ambiguous. Nevertheless, equity markets are affected asymmetrically. Th is also supports the idea of the reduction of financial contagion effect by imposing investment restrictions. To analyze the contagion implications, lets compute the correlation between domestic and foreign equity returns as follows: P = CTJCT//lkd|||k/||-We can rewrite the stocks' volatilities as follows: Sdad = S°do-°d + Be + 0{e2), Sfaf = S°fa°f -Be + 0(e2), with B = XS°f k w * P | | E ° - i ( t l + t 4 ) : ( ( l _ l ) ( l _ l o g ( ^ _ ( l _ | l o g 4 ) ) K ^ { ) Using this volatil ity expressions, the correlation is given as follows: P \S°Ja°J\ S°\\o-°\\J 1 -afB 0(e2). Table 4.2 presents simulated correlations between the domestic equity market returns and the foreign one. W i t h the set of parameters values we used, the correlation is lower than the benchmark one (p < p°) when the domestic country restricts its residents in their foreign investments. This results shows that, domestic country market w i l l be less affected by international shocks. The intuit ion for this is as follows. B y imposing investment barriers on its residents, domestic agents w i l l hold more of the supply of the domestic equity and also their wealth w i l l be essentially invested in home equity. Thus when shocks occur on the international stage, not only domestic stocks wi l l be less affected but also domestic agents' consumption w i l l be less impacted. W e l f a r e a n a l y s i s Similar to the previous welfare analysis, under the assumption of logarithmic pref-erences for the agents, the welfare of the agent is decomposed into two elements: Ui = U° + AUi, i e {d, / } , where U° is the welfare i n the bench-market and is given as follows U? = - ( l - e ^ T ) [ 2 1 o g ( ( l - e - ^ ) / / ? ^ ( 0 ) ) + l o g ( 5 ( 0 ) / , 5 * ( 0 ) ) ] / / 3 r rT rt +E [ e-* [ ( ^ ( r ) + ^ . ( r ) - (||a , ( r )|| 2 + ||a,.(r)|| 2)/2)drdt .Jo Jo and AUi is given in the following corollary. C o r o l l a r y 4 .5 .13. The welfare variation of agents are AUd 2(1 -E ) l og (X d (0 ) /X d ° (0 ) ) fe~* A l + 2A(r)) Jo Jo {[Sf(r)/S(r)-n] + Y | E - i ( r ) ( t l + l4)||a  A U f = + 2 ( l - e - ^ ) l o g ( X / ( 0 ) / X ; ( 0 ) ) t{[sf(T)/s(T)-*]+y drdt +E [ T e - » /V(r))2 Jo Jo | E - i ( r ) ( i l + i 4)||2 drdt Table 4.2: Simulated correlations between domestic and foreign equity returns. This table shows the simulated correlations between the domestic and foreign market re-turns. The values of the parameters used in the simulations are: T = 6 years, <$i(0) = <52(0) = <53(0) = 5*{0) = 100. In Panel (a), pSl = 0.03, us2 = u&3 = us* = 0.05, aSl = (0.25,0,0,0) T , aS2 = (0,0.25,0,0) T , ag3 = (0,0,0.25,0) T , cs* = (0,0,0,0.25) T . In Panel (b), uSl = us2 = us3 = ps* = 0.05, a5l = (0.15,0, -0.05, - 0 . 05 ) T , aSl = ( -0.05,0.15,0,-0.05) T , aS3 = (-0.05,0.05,0.15,-0.10) T , as. = (-0.05, -0.05, -0.05,0.15) T . Panel (a): Values of p° Values of p T- t 0.10 7T 0.30 0.50 T - t 0.10 7T 0.30 0.50 5 4 3 2 1 0.5755 0.5724 0.5735 0.5717 0.5736 0.5753 0.5758 0.5723 0.5742 0.5695 0.5716 0.5671 0.5647 0.5654 0.5663 5 4 3 2 1 0.2946 0.2639 0.2425 0.2082 0.1909 0.3883 0.3697 0.3403 0.3206 0.2908 0.4786 0.4555 0.4362 0.4209 0.4038 Values of p° Panel (b): Values of p T-t 0.10 7T 0.30 0.50 T - t 0.10 7T 0.30 0.50 5 4 3 2 1 0.6523 0.6558 0.6584 0.6592 0.6625 0.6495 0.6502 0.6528 0.6545 0.6515 0.6494 0.6503 0.6498 0.6497 0.6521 5 4 3 2 1 0.6372 0.6404 0.6427 0.6430 0.6460 0.6394 0.6398 0.6421 0.6434 0.6399 0.6448 0.6454 0.6446 0.6442 0.6465 . The welfare variation expressions in Corollary 4.5.13 have two components. The first component captures the variation of the in i t ia l wealth when domestic residents are re-stricted in their foreign investment. The second component represents the change in the hedging gain/loss relative to the benchmark holdings. Recal l , in the economy setting, we assume each agent to be endowed wi th the in i t ia l supply of his home country. From Corol -lary 4.5.12 above, since domestic stock value increases and foreign stock value decreases, the in i t ia l wealths X<j(0) increases and Xf(0) decreases from their benchmark levels. However, the drift of the changes in the relative proportion of domestic agent wealth in the economy is negative: Et [d{\(t)/(l + A(t))}] = Et [d{Xd(t)/(Xd(t) + Xf(t))}] < 0. Thus domestic agent wealth is decreasing, which implies that the positive wealth effect in his welfare quantification disappears over time. The hedging component of the welfare of domestic investors is decreasing while that of foreign investors is increasing. B y restricting its residents, the domestic government is making its citizens worse-off in terms of welfare. The welfare of domestic agents decreases and foreign agents are gaining wealth from the high expected excess return on their home stock. Indeed foreign agents invest more in home stock, and since foreign stock excess return is going up, they benefit from it . Also , foreign investors benefit from the increase in the domestic bond price run up. To sum up, imposing investment restriction on domestic residents' foreign investment creates at least two opposite effects. The first one (positive) is the reducing of the cost of capital for domestic firms, since domestic residents keep their capital home. The second effect (negative) is the welfare lost for the restricted domestic agents since they face l imited investment opportunity set. 4.6 C o n c l u s i o n Recent crises on international capital markets have reopened the debate on whether or not economic integration and market globalization are good for the countries involved. Whi le most of the international finance literature supports the idea that economic l ib -eralization is a worthwhile goal to pursue, the international macroeconomics literature presents a mixed view. Our a im in this paper is to provide policy makers wi th better guidance in their decision taking process. To achieve our goal, we develop a general equi-l ibr ium model of international capital markets in the presence of investment restrictions. We consider a continuous time, two-country, two-good exchange economy. Our main findings are as follow. First ly , when a domestic country caps foreign invest-ment in some key industries of the domestic economy, the cost of capital of the protected industry increases, and then through equilibrium, that of the non-protected industry decreases, al l else being equal. O n the other hand, when imposing restrictions on its residents' foreign investment, the domestic country improves its cost of capital. More-over, in both restricted economies, the domestic cost of risk free borrowing and lending is lowered. However, in the case of restriction on its residents' foreign investment, the uncovered interest rate parity relationship is violated. Secondly, by artificially restrict-ing agents' investment, countries can reduce the contagion effects on their market. Our results could explain why recent crises in international markets have had different effects on countries located in same geographical area or having similar economic characteris-tics. Third ly , when restrictions on investments are imposed, the effects on stock market volatilities are ambiguous. This result suggests that the debate on whether opening up financial markets is the cause of financial instability would not be fruitful at the general level because both more stable and less stable financial markets can occur in equil ibrium. Final ly , we show that when the restriction is protective, the welfare of the agents of the country imposing the restriction increases. Th is result helps us understand why some countries are so reluctant to change their protective financial policies. To our knowledge, this is the first attempt to l ink goods markets, financial markets and the real exchange rate in a dynamic international economy in the presence of invest-ment restrictions. In addition, our model offers tractabil ity without relying on numerical calculations. 4.7 Appendix Proof of Proposition 4.4.1 The representative agent uti l i ty is constructed as: U(5,5\ A) = max . f c ^ / ( l - 7) + c f " 7 ) / ( l - 7 )1 +A 1^/(1 - 7) + cf^/(l - 7) subject to Cd + Cf = 5 and c*d + c*j = 8*. The F O C s for this optimization are c/ = 8 — Cd, c*j = 5* — c*d, c^ 7 — X(8 — Cd)"1 = 0 and c * - 7 - \{5* - C ^ ) - T = 0, which yield cd = 8/(1 + A 1 / 7 ) , c*d = 5*/{l + A 1 / 7 ) , C / = 5 A 1 / 7 / ( l + A 1 / 7 ) , c) = 8*Xl^/(l + A 1 / 7 ) . Substituting back cd, c / ; c*d and c) in the ut i l i ty function expression yields: U(8, 8*, A) = ( l + A 1 / 7 ) 7 [ ^ / ( l - 7) + <5* ( 1- 7 )/(i - 7)]. The exchange rate is defined as the marginal rate of substitution of good 8* for good 8 and is given by: au(6{t),6'{t)1x)/ds' (m_y P { ) 8U(S(t),8*(t),X)/d8 \8*(t)J ' In the benchmark economy, A is constant. The equilibrium pricing kernel is the marginal ut i l i ty of the representative investor: m = __0tdU(8(t),8*(t),X)/d8 = ^ 1 -7 dU{5(0),8*(0),X)/d5 ~ [8(0) Apply ing Ito's formula to £(£), we get d^(t) = - e ( 0 ( ) 9 + 7 w W - 7 ( l + 7 ) l k * W l l V 2 ) d t - e ( 0 7 ^ ( t ) T d W ( t ) . Comparing this to the pricing kernel expression given in Equation (4.4.1), gives r(t) and 6(t). The risk premiums are obtained by p. — r = T.6. From the risk premium of the foreign bond, we obtain the interest rate differential. Stock prices are obtained by discounting future dividends using the economy pricing kernel. Constrained investor optimization problem Suppose that agent i is l imited to hold no more than TT of his wealth in stock j. To characterize the optimal policies, we use the framework developed by Cvi tanic and Karatzas (1992). Lets define the constraint space K of agent i: K = {TT such that TTJ < TT} , where TTJ is the percentage of wealth agent i invests in risky stock j, and 7f is the maximum percentage of wealth he can put on that stock. We define x = (XQ, X\, X%, X3)T and the support function of K, as T J TTXJ if xk = 0 and Xj > 0, k ^ j , 1p{3L) = S U p (7T X J = < ire K I oo otherwise. The dual cone K is then: K = {x, such that ip(x) < + 0 0 } = {x such that xk = 0 and Xj > 0, k ^ j}. Hence, V ' W = T T X J , for al l x e i f . The constrained optimization problem of agent i can be solved as an unconstrained one by designing a new financial market from the original market as follows: dBx(t) = (r{t) + ip(x(t)))Bx(t)dt dSx(t) = I%(t)(p(t)-x(t) + ip{xi{t)))dt + I%(t)Z(t)dW{t) Sx = (BXf, Sx, S f , SfV is the vector of the foreign bond and risky stocks, and I f = diag(Sx). The new (fictitious) pricing kernel facing agent i, ^ ( i ) , is: d£i(t) = [r,(£)d£ + ^(t) TdW^(t)] , w i th &(0) = 1, where = r + ^ ( x ) and 6i = 6 — S _ 1 x are the interest rate and market price for risk facing the constrained agent i. Having defined the fictitious market pricing kernel, agent maximization problem can be restated as the following variational static problem: max E f ' e - ^ [^ (4 ) ) + « « ( * ) ) ] dt Jo under the budget constraint: E f m [ci(t)+p(t)c*(t)] dt .Jo = Xi(0). To obtain the minimax pricing kernel, we have to minimize the equivalent dual prob-lem (see Cvitanic and Karatzas (1992) for the technical details). For that, lets calcu-late the dual functions of the uti l i ty functions u and v: u(y) = m a x z — zy and v(y) = m a x z v(z) — zy. The dual problem of agent i is then formulated as follows: mini? / e^luiyaty + viyPimmdt .Jo (4.7.1) Log utility case W i t h log util ity, the dual functions defined above become u(y) — v(y) = — (l+log(y)). From the dual optimization in Equation (4.7.1), x is obtained as follows x = argmin[2V'(x) + ||0 - E _ 1 x | | 2 ] . x = X2L3, where £3 = ( 0 , 0 , 1 , 0 ) T , x2 > 0. The optimization then becomes x2 = argmin[27f:E2 + \\0 - x 2 S _ 1 i 3 | | 2 ] . X2>0 The F O C of this optimization problem is: 7f + x 2 | | S - 1 t 3 | | 2 - ^ 2 - ^ 3 = 0, which implies X2 = W%¥[eTE~h3~*]+' (4'7,2) Proof of Proposition 4.5.2 Representative investor utility W h e n foreign investors are restricted, the representative agent ut i l i ty is constructed as: U{5,5*, A) = max m [u{cd) + v(c*d)} + A [u(cf) + v{c})] , subject to Cd + Cf = 5 and c*d + c*f = 5*. Solving this optimization yields, cd = 5/{l + \ l h ) , c*d = $ 7 ( 1 + A 1 / ? ) , C f = 5X^/(1 + A 1 / 7 ) , c*f = <5*A 1 / 7 / ( l + A 1 / 7 ) . Substituting back cd, Cf, c*d and c*f in the ut i l i ty function expression yields: U(5,6*, A) = ( l + A 1 ^ ) 7 [ ^ - T / ( I - 7 ) + 5*^/(1 - 7)]. E x c h a n g e r a t e The exchange rate is defined as the marginal rate of substitution of good 5* for good 5 and its process is given by: = dU(5(t),6*(t),\(t))/d6* = (S(t)y P { > dU(8(t),5*(t),\(t))/dS \8*(t)J • App ly ing Ito's lemma to the expression of p and comparing the terms to equation (4.3.1) gives the instantaneous return [iv and volati l ity av of changes in the exchange rate: Mp(t) = 7 (W(*) - W ( * ) ) - 7(1 + 7) ( I M * ) U 2 - |k«.(t)||2) 12 + 7 a , ( i ) T a p ( i ) , ° P ( * ) = 7(<7<5(*) - <?>(£)). T h e w e i g h t process The weight A is obtained as follows A(i) = u'(cd(t))/u'(cf(t)) = v'(cd(t))/v'(c}(t)) = ydMt)/yfSf(t), where is the lagrange multiplier of agent i's maximization problem, and & is the minimax pricing kernel of the fictitious market facing agent i described above in Appendix (4.7). App ly ing Ito's formula to the expression of A gives (dUt)dHf(t)) = Ht)[(rf(t) - rd(t) + ej(t)(9f(t) - ed(t)))dt + (9f(t) - 9d(t))TdW(t)}. Comparing this expression to the following process of A d\(t) = X(t)[fxx(t)dt + ax(t)dW(t)}, gives »x(t)=rf(t)-rd(t) + e](t)ax(t), and ^ ( t ) = of(t) - ed(t). From the constrained investor optimization in Appendix (4.7), since foreign agent is constrained in his portfolio holding, 77 = r + 7fx2 and Of = 9 — x2T,~1t3, where 7f is the l imit percentage holding, and i3 = (0,0,1,0) T. A n d since domestic agent has no investment restrictions, ^ d = £, so rd = r and 9d = 9. We can rewrite: lix{t) = Tt(t)x2{t)+9](t)ax{t), (4.7.3) ax(t) = 9{{t)-9(t) = -x2(t)Y1-\t)L3 or 6f(t) = 9{t) + ax{t). (4.7.4) Pricing kernel and optimal policies Following the same argument as Cuoco and He (1994), the pricing kernel is obtained as the marginal ut i l i ty of the representative agent: = dU(S(t),S*(t),X(t))/d8 _ 0 t / l + A ( t ) V n 7 (5(t)y ? U dU(6(0),5*{0),\(0))/d5 \l + \(0yh) \8(0)J ' App ly ing Ito's lemma to the expression of £ gives: - ( l ) - ( l ) - K 0 ) w To compute this expression, we first compute the following partial differentials: d<5 rq5' d82 n r)^82' d\ 1 + A V 7 ? ' d2^ ^ x i h - i x g2g 1 - 7 A i / 7 - 2 7i , \ i /~Sr ) Q \ O /-i , \ i / ^ \ 9 > ' dAd<5 ' l + A V 7 s 5 ' 5 A 2 7 (1 + AV7)2 V We use these part ial differentials in the expression of d£ above, which yields A 1 / 7 1 - 7 A 1 / 7 .. I l 2 A 1 / 7 T \ 1 + A V 7 ^ 27 ( l + A 1 ^ " ^ " '1 + AV7 / A 1 / 7 \ comparing this expression to dm = -£ (* ) [r(t)dt + 6(t)TdW(t)] , we get r ( i ) = /5 + 7 M ^ ) - 7 ( l + 7 ) l k ^ ) | | 2 / 2 Using the expressions of / i A and CTA from Equations (4.7.3)-(4.7.4), r and 0 simplify as follows: r(t) = /3 + 7 W ( t ) - 7 ( l + 7 ) | | «7 S ( t )|| 2 /2 A ( * ) 1 / 7 - 1 + 7 A(t)V7 , ^ . . ^ ^ , , 2 7f(i)x2(t) - ' ' , \ ' J ^ W ) 2 ! ! ^ 1 ^ ! ! 2 , l + A(t)V7 2 7 (1 + A(£)V7)2 A M 1 / 7 0(f) = ^as(t)+ x2E-x(t)L3. The risk premia are obtained by using p — r = £0. From the risk premium of the foreign bond, the difference between the two countries risk-free rates is obtained as follows: r* + i i p — r — <7T0. Proof of lemma 4.5.3 Stock prices are obtained by discounting future dividends using the pricing kernel Sj(t) =E,' where £ is the pricing kernel. We denote by D t the Mal l iav in derivative operator at time t. Assuming a l l the regularity conditions satisfied for the different Mal l i av in derivatives to exist, we first compute the following Mal l i av in derivatives using Mal l iav in chain rule: D t ( £ 0 t o ( a ) ) = ^ ( S ) D t £ ( S ) + £ ( 5 ) D ^ . ( S ) ; D t £(s ) = ^ D t A ( s ) + ^ D t < J ( s ) and D t £( i ) = - 0 ( f ) £ ( i ) . We now take the Mal l i av in derivative of Sj D t £(t ) D ^ ' W = - ^ - E t [ j t Bt(C(s)5j(s))d& 'j\(s)BtSj(s)d£ « * ) r W)Et LJt T 5j(s)^-Bt5(s)ds Comparing this Mal l i av in derivative to the following Mal l iav in derivative of Sj, sy c gives crj(i). Proof of Corollary 4.5.4 The proof follows by applying Ito's lemma to the expressions Q = 5/(1 + A 1 / 7 ) , C f = 5X^/(1 + A V T ) , = 6*/(l + A 1 / 7 ) , and c} = <5*A 1 / 7 /(l + A 1 / 7 ) . Proof of Corollary 4.5.5 From Equat ion (4.7.2), x2 = in the expression of x2, we get x2 = • „ i „ 2 [ 0 T S - 1 t 3 - 7 r ] + . Using 0 = a 5 + - ^ - x 2 S - 1 t 3 1 + A [ajE-h3-n]-i l S - ^ a l l 2 with i 3 = (0, 0 , 1 , 0 ) T and xo = x\ = x3 = 0. In the expression of x2, n represents the percentage wealth holding threshold in stock 2. We need to establish the equivalence between the percentage stock holdings and the percentage wealth invested in the stock. Suppose Ttw is the maximum percentage of wealth the foreign investor can invest in the domestic stock 2, and 7fs the maximum percentage of shares of stock 2, foreign investors can hold. We w i l l show that there is a relationship between TTW and 7fs. That allows us to use the results from Appendix 4.7. The relationship is as follows: nwXf = TTSS2, and implies TTw = itsS2/Xf. Lets denote by S the total wealth of the world, it is given as follows: S = Xj, + Xf = S\ + S2 + Sf. If there is no constraint, the percentage of wealth foreign agent wi l l invest in stock 2 is S2/S, and the percentage shares of stock 2 he w i l l hold w i l l be: A / ( l + A). The wealth of foreign agent is Xf = SX/(1 + A) and that of domestic agent is Xd = 5 / (1 + A). Thus S2 SS21 + X „ '2 _s- (4.7.7) Using Equat ion (4.7.7) in the expression of x2, we get %2 1 + A , T 1 1 + 1 + A „ T V - I , S 2 1 + A ,|s-^ 3 || 2 0 J ||s - i i 3 || 2 Since we are in a log preference setting, S(t) = 2(1 — e~^( r _^)(5(i)//?, hence its volati l ity is as- B u t since S = Si + S2 + S / , its volatil ity is also the weight average of the three volatilities as follows: as = (Si/S)ai + (S2/S)a2 + (Sf/S)af. Also recall £ = (<jp, <7i, a2, o"/) T. So c r j E _ 1 i 3 = S2/S, and x 2 is simplified as follows x2 1 + A ^ T V - 1 , - s $2 1 + A 1 + A 5 2 [ A / ( l + A) - 7r']H | E - ^ 3 | | 2 5 A / ( l + A) If there is no constraint, the percentage shares of stock 2 foreign agent w i l l hold w i l l be: A / ( l + A). The drift and volati l ity of the weight process become fix(t) = 7tw(t)x2(t) + 9](t)ax(t), 1 + X(t) 5 2 (t ) X(t) S(t) ' = ^ ( ^ ^ 2 W - 2 ( ^ J ( ^ - ( ^ 3 + T ^ | | S - 1 ( ^ 3 | | 2 , = 0, <T\(t) = - X 2 ( t ) S _ 1 ( t ) t 3 -Using Equat ion (4.7.5) and = 0, <7j£_1i3 — S2/S, the risk free rate becomes r = (5 +^ - \\a5\\2 + OA = P + M<5 - || 0"51| - T — T - ^ ^ -1 + A 1 + A 5 Equat ion (4.7.6) gives the Sharpe ratio 0 = 05 + Y ^ X 2 Z 1t3. P r o o f o f C o r o l l a r y 4.5.6 Stocks prices are computed using perturbation methods. We follow three steps. Step 1: we compute the equity prices S<i and Sf. Step 2: we compute the prices Si and 5 2 using results from step 1. A n d lastly in Step 3: we compute the volatilities of the stocks and the volati l ity of the weight using results from Step 1 and Step 2. Step 1 Discounting future dividends using the pricing kernel, the foreign country stock index is Sf(t) = 5(t)Et e - 0 i s - t ) i ± m d s 1 + A(t) Since A is a martingale, it is easy to see that + 6(t)Et -P(s-t) t ) l + X(s) 63(s) 1 + A(i) 5(s) ds -y — g—/3(T—t) S m t f e - K - V f f i d s 5 /W = a *(*) + + T12T^H{X(t),83(t)/S(t),t), (3 ' 1 + A(i) 1 + A(t) where H(X(t),83(t)/S(t),t) = Et [ j f e-^X(s)s-^lds\ and H(X(T), 53(T)/5(T),T) = 0. In absence of constraints Sf(t) = Sf(t). The total market capitalization of the world is S(t) = + S/(t) = 2(1 — ~P(T-t) )5(t)/(3, which implies Sd(t) = 2(1 -e-^)5(t)/(3-Sf(t). Assuming d(S3/8) = (53/S)[ps3/6dt + os3/sdW(t)}, w i th n$3/$ n o n stochastic, then l - e - P ( T - t ) fr A o m Amp/5* Sf(t) = t). App ly ing Ito's lemma to the expressions of and Sf(t) give l - e - « r - ' U ( i ) , , ^ ( i ) / , T e / ; H + M s ' M T ) ) < ' T ^ / N *'<*> = ^ t ) + g,(t)(l + A(t)) a " { t ) / e/3t5(t)Hx e*5(t)H 83{t) £ e ^ + ^ ) ) d r d + \Sf{t)(l + A(t)) 5/(t)(l + A(£)) 2 5 , (0 (1 + A( i ) ) 2 J 1 J M j +TTW)sW)s{) Sf(t)(i + A(t)) ( 4 7 " 8 ) _ [2(1 - e - ^ ) ) c 5 ( £ ) / / 3 ] a 3 ( Q - S ^ f f l  d W 2(\-e-KT-*))8{t)/p-Sf{t) " 1 j We write H as & series of e. W h e n e = 0 the problem reduces to the bench-market case. We choose e = ^ _^  • The Taylor expansion of i 7 wi th respect to e is H = H° + Z%=1Hnen + 0 ( e w + 1 ) where = dnH/den(e = 0) for n = 1 , N and = H(e = 0). Under appropriate regularity conditions, it can be shown that Z(t)Sf(t)+ f\(s)(83(s)+p(s)8*(s))ds Jo is a martingale under V (see Cuoco & He (1994), and Basak & Gallmeyer (2002)). Therefore, the drift must be zero. Hence, H solves the quasi-linear partial differential equation ( £ + §-t)H(X(t),83(t)/8(t),t) + e-0t\(t)83(t)/8(t) = 0 with boundary condition H(\(T),83{T)/S(T),T) = 0. We solve for H°(\(t),S3(t)/8(t),t) = \(t)Et \ [\-e°5-^ds] = Xit^e-V f e ^ ^ ' ^ d s , and for Hn, n > 1, by solving the following part ial differential equation J)2 Hn i / A . \ 2 ffi un ^ II <T\ II A > 8 3 ^ d2Hn „ . 53^ OH 1„ 1 , 2 , 2 ^ ^ " „ l , i 112/^3 \ V - d2Hn n with Hn(X(T),83{T)/8{T),T) = 0. Lets denote by g = <S3/<*- P u t t i n g together the e terms, the part ial differential equation for H1 is: ^ I K I l V ^ + pggHg = ffTE^i4s^l + A L | 5 f l o v ( 4 . 7 . 1 0 ) For ease of exposition and also in order to provide intuitive economic interpretations for the derived formulas to follow, we assume \ig = LIS3/S = 0 . We solve the P D E ( 4 . 7 . 1 0 ) to obtain H1. Also recall, CTJS0-1^ = S%/S and afTTli3 = 0. From a°f = J o a 5 + 1 ^ " ' ^ 3 ^ 3 , we compute a J 3 E 0 h3 = - 2 ( 1 _ e - g f l - 0 ) f e / j r So, cr]Z°-h3 2(l-e-P(T-t))53/f3 S Hence equity prices are A S xs S! S 0 ° ( l+ 5 ) ( l - log( 5 ) )e , Next we compute the individual stock prices in the domestic market. S t e p 2 The prices of domestic stocks are S2{t) = S(t)Et c - g ( s _ t ) l + A(s) 82(8) d s 1 + X(t) 5(s) 5i(t ) = Sd(t)-S2(t), where h(\(t),62(t)/5(t),t) = Et [f e - * A ( » f g > d s ] , and h(X(T), 82(T)/S(T),T) = 0. In absence of constraints = S?(t). A p p l y i n g Ito's lemma to the expressions of Sj gives S°2(t)a°2(t) + e^8(t)has{t) + e^8(t) J ^ J l ^ ^ t ) + e d(<V*) S(t) ^S(t)^ - 5 2 ( i ) ) A ( i ) ^ W ] / ( l + A(t) )S a (t ) , ^ ) + c i 2 ( l , M t ) - ^ W ) . & ( t ) - S 2 ( i ) We write / i as the Taylor series expansion of e: h = h° + Z%=1hnen + 0(eN+1) where hn = dnh/den(e = 0 ) for n = 1 , A T and /i° = /i(e = 0 ) . Aga in under appropriate regularity conditions, it can be shown that ( 4 . 7 . 1 1 ) ( 4 . 7 . 1 2 ) £ ( t ) S 2 ( t ) + [tas)82(s)ds Jo is a martingale under V (see Cuoco & He (1994), and Basak k Gallmeyer (2002)). Therefore, the drift must be zero. Hence, h solves the quasi-linear partial dif-ferential equation (£ + j^) h(\, 62/8, t) + e~^X(t)j^- = 0 w i th boundary condition h(X{T),S2(T)/S(T),T) = 0. We solve for h°(X(t),S2(t)/8(t),t) = e-^X(t)^-, and for hn(X,82/5,t) by solving the part ial differential equation 82hn 1, dx2e +^s2/s\ i) £ d2hn 2un 0"v <5 J ^ d(82/5)2 n 2 ^ d2hn n 52^ dhn n n with hn(X(T),52(T)/5(T),T) = 0. For ease of exposition and in order to have interpretable approximate expressions, we assume Hs2/s = 0. We solve the above P D E to obtain h1. We can also show that: (a§2 — <7<5)TE0-1i3 = 1 — S2/S. Hence, domestic stock prices are 2A So — So 5i = S° + \\'S2/5 XS r 2 ( l - f ) ( l - l o g ( f ) ) (l + f ) ( l - l o g ( f ) ) e. Step 3 The last step consists of using the stock prices to obtain the volatilities. We use 4.7.8, 4.7.9, 4.7.11 and 4.7.12 and solve the system of simultaneous equations to get the volatilities of the stocks. P r o o f o f C o r o l l a r y 4.5.7 Agent i 's welfare is given as follows: Ui = E / V * [log(C i(t)) + log(c*(i))]di vo wi th a(t) = e-P/y&it) and c*(i) = e~0t/yiP(t)^{t) where V l = (1 -e~l3T)/pXi(0) and Xi(0) is agent i 's in i t ia l wealth. Agents' wealths are: Xd{t) = 2 ( i _ e - /J (r - t ) ) e-V/pydUt) and Xf(t) = 2 ( l - e ~ ^ ) / Pyf£f(t). The fictitious pricing kernel facing agent i (Appendix (4.7)) is The welfare function then becomes rT U = E E e-fit [-2(3t - 21og(2/0 - log(&(*)) - log(p(t)&(*))] dt LJO [ e-*31 \-2(5t - 2 logfo) - log(p(t))] dt] - 2£7 1 7 e""' logfe .Jo J Lvo i(t))dt We can also rewrite rt and ^ as follows: = f3+(j,s — \\os\\2+Ari(X) and = a^ + A ^ ^ A ) , where Ar^(A) and A0j(A) represent the deviation from the benchmark quantities. A n d then using these expressions and the process of p, we get r rT E +E +2E [ e - ^ ( - 2 1 o g ( y i ) - l o g ( p ( 0 ) ) ) d i Jo f e-^ f {ns[r) + Mr) ~ (\WS(T)\\2 + \\OS.(T)\\2)/2) drdt Jo Jo f e-P* f (Ar , (A(r ) ) + as[rf M ^ T ) ) + ||A^(A(-r))||2/2) drdt Jo Jo or Ui [ e-* (21og((l - e-^/pxm + log(<J(0)/<J*(0))) dt Jo +E \ f f ( W ( r ) + ^ . ( r ) - (|Mr)|| 2 + ||^(r)||2)/2) drdt Jo Jo fT e-* f (Ar i (A ( r ) ) + a , ( r ) T A ^ ( A ( r ) ) + ||A^(A(r))||2/2) di<S£ Jo Jo +2E .13) To obtain the final expressions, we replace Ar^(A) and A#j(A) by their respective expres-sions depending on the restriction case. The expressions of A V J ( A ) and A#;(A) are: for domestic residents A m = A ( Q (S2(t)\2[X(t)/(l + X(t))-n(t)]+ A ^ A J | | E - i ( i M 2 U ( * ) ; A( i ) / (1 + A(t)) ' Af> m A(t) S2(t)[X(t)/(l + X(t))-7t(t)}+^-lny. M d { X ) - W H t ^ m A(t) /(1 + X(t)) s W t 3 ' and for foreign agent Remark 4.7.1. Recall, in this case Tt = ns is the limit percentage shareholdings of stock 2 not the percentage of wealth invested in stock 2. Thus we use Equation (4.7.7). The expressions of AT~J and A#j are replaced in {/d and Uf. Constrained investor optimization problem Suppose that agent i is l imited to hold no more than ff of his wealth in securities j and k together. To characterize the optimal policies, we use the framework developed by Cvi tanic and Karatzas (1992). Lets define the constraint space K of agent i: K = {n such that 7Tj + Kk < TT, 3' ^ k} > where TTJ is the percentage of wealth agent i invests in security j, and n is the maximum percentage of wealth he can put. We define x = (xi)t and the support function of K, as T I irx if Xi = 0 and Xi = = x > 0, l=£k,j, yj(x) = sup ( i x ) = < TT6 K [ co otherwise. The dual cone K is then: K = {x, such that ^(x) < + 0 0 } = {x such that Xj = xk = x > 0 and xi = 0, / 7^  j , A;}. Hence, -i/>(x) — ^x, for al l x e l This constrained optimization problem can be solved as an unconstrained one by designing a new financial market from the original market as follows: dBx{i) = (r(t) + ip(x(t)))Bx(t)dt dSx(t) = I^{t)(iJ,(t)-yi(t) + ^(yi(t)))dt + I^{t)E(t)dW(t) Sx = (Bj, Sf, S f , Sj)r is the vector of foreign bond and risky stocks, and J J = diag(Sx). The new (fictitious) pricing kernel facing agent i, &(£), is: <%i(t) = -e<(t) [n(t)dt + 9i{t)TdW{t)], wi th 6(0) = i , where rt = r + ^(x) and 9i = 9 — S _ 1 x are the interest rate and market price for risk facing the constrained agent i. Having defined the fictitious market pricing kernel, agent maximization problem can be restated as the following variational static problem: r rT max E / °i'Ci [JO under the budget constraint: -pt -u(a(t)) + v(c*(t))}dt E Jo = ^ ( 0 ) . To obtain the minimax pricing kernel, we have to minimize the equivalent dual prob-lem (see Cvi tanic and Karatzas (1992) for the technical details). For that, lets calcu-late the dual functions of the uti l i ty functions u and v: u(y) = m a x 2 u(z) — zy and v(y) = m a x z v(z) — zy. The dual problem of agent i is then formulated as follows: rT mini? / e-^[u(yat)) + v(yp(t)m)}dt Jo (4.7.14) L o g u t i l i t y case W i t h log util ity, the dual functions denned above become u(y) = v(y) = —(l+log(y)). From the dual optimization in Equation (4.7.14), x is obtained as follows x = argmirr[2V'(x) + \\9 - S _ 1 x|| 2 ] . x€K x = x(ti + ii), where tj = (.., 0 ,1,0, . . ) T is a vector of zero everywhere and one in the jth row, XQ = %z = x > 0 and xi = 0 for / ^ 0,3. The optimization then becomes x = argmin[27ra; + — xT, + 4^) 112] -x>0 The F O C of the optimization problem is: 7f + x H S " 1 ^ ! + i 4)|| 2 - 0 T S _ 1 ( t i + H) = 0, (4.7.15) which implies x = x0 = x3= | | E , 1 ( ^ + 4 4 ) | | 2 [ ^ E - 1 ^ + , 4) - 7 f ] + . Proof of Proposition 4.5.9 Representative investor utility W h e n domestic residents are constrained, the representative agent ut i l i ty is con-structed as: U(6, 6*, X) = max \u(cf) + v(c})] + X[u{cd) + v(c*d)] , Cdfij ,c*d,c*f subject to cd + Cf = 6 and c*d + = 5*. Solving this optimization yields cf = 5/(l + A 1 / 7 ) , c) = 5*/{l + A 1 / 7 ) , c d = 5X^/(1 + A 1 / 7 ) , c*d = 5 * A 1 / 7 / ( l + A 1 / 7 ) . Substituting back cd, Cf, c*d and c*f in the ut i l i ty function expression yields: U(6,5*,X) = ( l + A 1 / 7 ) 7 [ ^ / ( l - 7) + ^ ^ " ^ / ( l - 7)]. Exchange rate The exchange rate is defined as the marginal rate of substitution of good 5* for good 5 and its process is given by: dU(5(t),8*(t),X(t))/d5* (6(t)Y> P { ) dU(5(t),5*(t),X(t))/d5 \S*(t)J • Apply ing Ito's lemma to the expression of p and comparing the terms to equation (4.3.1) gives the instantaneous return \iv and volati l ity ap of changes in the exchange rate: M * ) = 7 ( W W - M * ) ) - 7 ( 1 + 7) ( I M i ) | | 2 - ||M*)||2) / 2 + 7^(t ) T cr p (£ ) , ffp(t) = 7(o-c5(i) - o>(*))-T h e weight process The weight A is obtained as follows X(t) = u'(cf(t))/u'(cd(t)) = U ' ( c } ( £ ) ) /^ (^ (£ ) ) = V&{t)IVdUt), where y_i is the lagrange multiplier of agent i's maximization problem, and £j is the minimax pricing kernel of the fictitious market facing agent i described above in the Appendix 4.7. App ly ing Ito's formula to the expression of A gives = A(t)[(r a(t) - r , ( t ) + «J(()(«a(t) " « , ( * ) ) ) * + ( W O - 6,{t)fdW(t)}. Comparing this expression to the following process of A dX(t) = \(t)[nx(t)dt + ax(t)dW(t)\, gives fiX(t)=rd(t)-rf(t) + 9J(t)ax(t) and ax{t) = ed(t)-ef{t). From the constrained investor optimization in Appendix (4.7), since domestic agent is constrained in his portfolio holding, rd = r + TTX and 9d = 6 — x E _ 1 ( i ! + 14), where t\ = ( 1 , 0 , 0 , 0 ) T , ii = (0 ,0 ,0 ,1 ) T and TT is the l imit percentage holding. A n d since domestic agent has no investment restrictions, = £, so 77 = r, 9f = 9. We can rewrite: fix(t) = nx(t) + 0j(t)ax{t), (4.7.16) o-x(t) = 0d(t)-9{t) = -x{t)Z-1(t)(L1 + t4) or 9d(t) = 9(t) + ax(t). (4.7.17) Pricing kernel and optimal policies Following the same argument as Cuoco and He (1994), the pricing kernel is obtained as the marginal ut i l i ty of the representative agent _gt dU(5(t),5*(t),\(t))/d6 _ &t (1 + A ( t ) V 7 y f5{t) ? W dU(5{0),5*(0),\(0))/d5 yi + X{0y/-rJ \5{0) App ly ing Ito's lemma to the expression of £ gives: (I)-(S)<-)4(§)^ To compute this expression, we first compute the following part ial differentials: d£ _ 1 8% 1_ 85 7 V 852 ~ 1 [ ^82' 8£ _ A 1 / 7 " 1 8X ~ TTAVT" 82j A 1 / 7 - 1 1_ 9 2 e _ 1 - 7 A 1 / 7 ' 2 ~ 7 l + A 1 / 7 ^ ( 5 ; ,9A2 ~ 7 ( l + A 1 ^ ) 2 ^ ' We use these partial differentials in the expression of <i£ above, which yields dm A 1 / 7 1 + A V T + \lo-s -(/3 + 7 ^ - ^ f ^ l K | | 2 MA 1 - 7 A x / 7 27 ( A V 7 A I / 7 1 + AV7 comparing this expression to d£(t) = -£{t) [r(t)dt + 6{t)TdW{t)} , we get r(t) P + ^6(t)-j(l + 1)\\as(t)\\2/2 A ( i )V7 + 9(t) = -yas(t) 1 + A(t)V7 A ( i ) V 7 7a [ 5 ( i ) T <r A ( i ) - nx(t) Mt)-1 - 7 I K 2 7 1 + A ( i ) 1/7 (4.7.18) (4.7.19) 1 + A(i)V7 Using the expressions of [i\ and o\ from Equations (4.7.16)-(4.7.17), r and 9 simplify as follows: . r(t) = /3 + 7 ^ ( t ) - 7 ( l + 7)||a,(i)||2/2 The risk premia are obtained by using p, — r = £#. The difference between the two countries risk-free rates is obtained by the risk premium expression on foreign bond as follows: r* + /J,p — r = aj9. Proof of Corollary 4.5.10 The proof follows by applying Ito's lemma to the agents' consumption expressions cd = SXl/y{l + A 1 / 7 ) , C / = 5/(1 + A 1/7) ; c* = s*\lh/(l + A 1 / 7 ) , and c) = 5*/(l + A 1 / 7 ) . Proof of Corollary 4.5.11 Using Equat ion (4.7.15), and 9 = as + S~1(<-i + IA)X, the F O C becomes 1 + A vf + ^ - L - x H E - 1 ^ ! + t4)||2 - ffjE"1^! + i 4 ) = 0, (4.7.20) which implies The volati l ity of the weight process is ax = - x E _ 1 ( i i + i 4 ) , and the drift of the weight process becomes MA 7rx + 9jax = irx + ^crg - ^ ^  ^ E 1 ( t i + t 4 ) ^ ( - E + t 4 )x ) , = Tfx - ffjE-.^ii + i 4 ) x + — J — - | | E _ 1 ( n + t 4 ) | | V , 1 + A = a^Tf - c r j E " 1 ^ ! + t 4 ) + 7 ^ l | S - 1 ( t i + t 4)|| 2x). Using the F O C above, we can see that p,\ — 0. Using the riskfree rate expression (4.7.18), and p\ = 0, the risk free rate becomes r = /3 + ^ - | N I 2 + y^a]ax = f3 + ps-\\o-s\\2-j^xa]Z-1(L1 + L4). Since we are in a log preference setting, the total wealth available in the world is S(t) = S1 + S2 + Sf = 2(1 - e-KT-Q)5(t)/p, hence its volati l ity is the weight average as follows: as = (S\/S)ai + (S2/S)a2 + {Sf/S)af. Also recall E = (ap, a\, a2, <7/)T. So a j E - 1 t i = 0 and < r j E ~ V = Sf/S. Therefore the risk free rate is r = [3 +us- \\crs\\2- Y^-jxSf/S. 110 From Equat ion (4.7.19), the Sharpe ratio is 0 = 0-5 + Y ^ X Z +t4). Proof of Corollary 4.5.12 Stocks prices are computed using perturbation methods. We follow three steps. Step 1: we compute the equity prices Sd and Sf. Step 2: we compute the prices S i and S 2 using results from step 1. A n d lastly in Step 3: we compute the volatilities of the stocks and the volati l ity of the weight using results from Step 1 and Step 2. Step 1 . Discounting future dividends using the pricing kernel, the foreign country stock index is Sf(t) = 5(t)Et T - g ( a - 0 l + A(a) 1 + X(t) ds + 5(t)Et c_0la_t)l + \(s) 83(s)ds 1 + X(t) 5(s) Since A is a martingale, it is easy to see that sf(t) 1 _ e-0{T-t) 6(t) + 5(t)Et\£e-^5-§$ds + P±^H(X(t)Mt)/S(t),t), P ' 1 + A(i) 1 + A(t) where H{X{t),53{t)/6{t),t) = Et [ j f e-^X(s)^ds\ and H(X(T),S3(T)/S(T),T) = 0. In absence of constraints Sf(t) = SJ(t). The total market capitalization of the world is S(t) = Sd(t) + Sf(t) = 2(1 — e-ftT-Q)5(t)/p, which implies Sd(t) = 2(1 -e-^)5(t)/P-Sf(t). f Assuming d(83/6) = (83/5)[/j,s3/sdt + as3/sdW(t)], w i th /j,g3/s non stochastic, then 1 _ e-P(T-t) rT Sf(t) = P (t)/5(t),t). App ly ing Ito's lemma to the expressions of Sd(t) and Sf(t) give / e^S(t)Hx e^5{t)H 53(t) J t T e ^ - / ^ 3 / ^ ) ) ^ , + + A(t)) + A(i))2 S/(t)(l + A(t))2 J A ( i ) C T A ( f ) I e*H S(t) e(*53(t)HS3/5 +TTW)W)si) sf(t)(i + \(t))as*/5{t)> ( 4 J ' 2 1 ) „(t\ - [W-e-W-*))6(t)/0\vs(t)-Sf(t)*f{t)  d { ) ~ 2(1 -e-f>(T-*))5(t)/{3- Sf(t) ' [ 4 J - 2 2 ) We write H as & series of e. W h e n e = 0 the problem reduces to the bench-market case. We choose e = [Sj/§ — 7f ] + . The Taylor expansion of H w i th respect to e is H = H° + Z%=1Hnen + 0{eN+1) where Hn = dnH/den(e = 0) for n = 1 , J V and # ° = H(e = 0). Under appropriate regularity conditions, it can be shown that mSf(t) + [\{s)(63(s)+p(s)5*(s))ds Jo is a martingale under V (see Cuoco & He (1994), and Basak & Gallmeyer (2002)). Therefore, the drift must be zero. Hence, H solves the quasi-linear partial differential equation ( £ + ft)H(X{t),53(t)/S(t),t) + e~0tX(t)63(t)/5(t) = 0 wi th boundary condition H(X{T),53{T)/6(T),T) = 0. We solve for H»(X(t),53(t)/5(t),t) = X(t)Et St3(-P+PS3/6(T))dTds and for Hn, n > 1, by solving the following part ial differential equation &un 1 /x_\2 d2Hn with F n ( A ( r ) , 6 3 ( T ) / 5 ( T ) , T ) = 0. Lets denote by g = <J3/<y. P u t t i n g together the e terms, the partial differential equation for H1 is: \\W9\\292Hlg + WH] = a j E - ^ p ^ ^ ^ . • (4.7.23) For ease of exposition and also in order to provide intuitive economic interpretations for the derived formulas to follow, we assume [ig = fj,g3/s = 0. We solve the P D E (4.7.23) to obtain H1. Also recall, ajT,0'1^ + t 4 ) = S°f/S and c r ^ E 0 " 1 ^ + t 4 ) = 1. . So, From a°f = ^zas + —j^s—-S3as3, we compute CFJJI0 + t 4 og E 0 - 1 ^ ! + t 4 ) = 2(i-e-^-t))Sa/p ~ -f- Equi ty prices are 1l 2(i-e-5(T-()) 5 3 / / 3-Sf = 5 2 -5 d — + A5? |a s || 2 ||E-i ( , 1 + , 4) k 9 |H|E°- i ( t l + t 4)|p ? ( l - 5 ) ( l - l o g ( 5 ) ) e , ( l - p ) ( l - l o g ( p ) ) e . Next we compute the individual stock prices in the domestic market. S t e p 2 The prices of domestic stocks are 5 2 ( i ) = 5{t)Et c _ 8 ( s _ t ) l + X(s)52(s)d^ 1 + A(i) 8(s) i f i o + ^ ( A ( t ) ' * » ( t ) / w ' t ) ' = Sd(t)-S2(t), where /i(A(t),5 2(t)/r5(t),t) = £ t [ f / V ^ A ^ f ^ - i s ] , and fc(A(r),$2(r)/<J(T),r) = 0. In absence of constraints Sj(£) = S°(t). App ly ing Ito's lemma to the expressions of Sj gives o2(t) ax{t) (f5{t)^ ~ S2(t))\(t)ax(t)] /(l + A(i))5 2(i), + M * ) - <r2(t)). Sd(t) - S2(t) We write h as the Taylor series expansion of e: / i = h° + E ^ L ^ e " + 0 ( e 7 V + 1 ) where hn = dnh/den(e = 0) for n = 1 , / V and /i° = /i(e = 0). Aga in under appropriate regularity conditions, it can be shown that (4.7.24) (4.7.25) at)S2(t)+ f Z{s)52{s)ds Jo is a martingale under V (see Cuoco & He (1994), and Basak & Gallmeyer (2002)). Therefore, the drift must be zero. Hence, h solves the quasi-linear part ial dif-ferential equation ( £ + -|) h(X, 52/S, t) + e - / 3 t A ( £ ) ^ ^ = 0 wi th boundary condition h(\(T),62(T)/6(T),T) = 0. We solve for For ease of exposition and in order to have interpretable approximate expressions, we assume fj,g2/g = 0. We solve the above P D E to obtain h1. We can also show that: (ag2 — <7<5)T£°~1i3 = 1 — S2/S. Hence, domestic stock prices are The last step consists of using the stock prices to obtain the volatilities. We use 4.7.21, 4.7.22, 4.7.24 and 4.7.25 and solve the system of simultaneous equations to get the volatilities of the stocks. P r o o f o f C o r o l l a r y 4.5.13 The proof is similar to the proof of Corollary 4.5.7. Therefore we can use the expres-sion of Ui in Equat ion (4.7.13). The final expressions are obtained by replacing Ar^(A) and A#j(A) by their expressions as follows. For foreign agent wi th hn(X(T),S2(T)/5(T),T) = 0. S t e p 3 Arf(X) X aJE \ti + t4)x, A6f(X) 1 + A A , E " 1 (<4 + <-4), X 1 + A Using these expressions, A , T v . - l / . , , \ , „ A T v - l / Arf + ajA9f + \\Mf\\2/2 = ~XYT\ 6 ( t l + L a ) + X\ + \ 6 ( t l + L i ) + x 2 | | r ^ E - 1 ( , 1 + t 4)||2/2 For domestic agent A + A „ _ , , , 1 Ard(X) = -x——rcrs £ l(ii+ LA) + TTX 1  A A6d(\) = X—-X-1(L1+L4)-xZ-1(L1 + LA) = -X-—Z-I(L1 + LI). 1 + A 1 + A Using these expressions, Ard + aj A8d + \\A6d\\2/2 = -x—X— ajYT1^ + t 4) + TTX 1 + A - x ^ a j E " 1 ^ + t 4 ) + X ^ J L E " 1 ^ + , 4)|| 2/2, = x 2 | | — ^ — £ _ 1 ( i i + i 4)|| 2/2 - x t T j £ _ 1 ( t i + t 4 ) + Tfx. 1 + A M u l t i p l y i n g the F O C (4.7.20) by x yields xajYT1^ + t 4) - TTX = ^ ^ - L - I I S - 1 ^ + i 4 ) | | 2 . Thus, Ard + a]Add + \\Add\\2/2 = - ^ ^ I I E " 1 ^ + t 4 ) | | 2 + x 2 ^ ^ | | E - 1 ( t 1 + i 4 ) | | 2 Bibliography [1] Basak, S. and B . 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