UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Essays in corporate finance Colpitts, Jeffrey Charles 2007

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-ubc_2007-266991.pdf [ 3.87MB ]
Metadata
JSON: 831-1.0100579.json
JSON-LD: 831-1.0100579-ld.json
RDF/XML (Pretty): 831-1.0100579-rdf.xml
RDF/JSON: 831-1.0100579-rdf.json
Turtle: 831-1.0100579-turtle.txt
N-Triples: 831-1.0100579-rdf-ntriples.txt
Original Record: 831-1.0100579-source.json
Full Text
831-1.0100579-fulltext.txt
Citation
831-1.0100579.ris

Full Text

ESSAYS IN CORPORATE FINANCE By JEFFREY CHARLES COLPITIS B.B.A., Bishop's University, 1999 M.Sc. Business, The University of  British Columbia, 2002 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In THE FACULTY OF GRADUATE STUDIES BUSINESS ADMINISTRATION THE UNIVERSITY OF BRITISH COLUMBIA March 2007 © Jeffrey  Charles Colpitts, 2007 Abstract In the first  essay, I consider the impact of  tort liability on firms  capital structure. Tort litigation is not only a substantial risk facing  firms  worldwide, but is also a unique form of  risk, in that it can be exacerbated or mitigated by how firms  adjust their debt-equity mix. I examine how firms  ought to adjust their capital structure when faced  with litigation, and consider various extensions to basic model. These include the interaction between capital structure, tort liability and insurance, how the problem changes when several firms  face  tort risk and are jointly and severally liable, and the implications that arise from moving from a one period to a two period setting. In the second essay, we develop and test a theory of  insurers' choice of the mix of  equity and liabilities. The role of  equity in insurance markets and in our model is to back insurers' promises to pay claims when there is aggregate uncertainty, or dependence among risks. Depending on the nature of  this aggregate uncertainty, the equity held by firms  in a competitive insurance market may increase with rising uncertainty, or it may initially increase then decrease. The ratio of  equity to revenue unambiguously increases with uncertainty. We test the model, as well as implications of  recent models of  insurance market dynamics, on a cross-section of  U.S. property-liability insurers. In the third essay, I examine optimal contracting with risk averse managers. I start from the following  observations: (1) managers select projects and exert effort;  (2) risk averse managers make distorted project selection decisions, and this problem is increasing in risk aversion; (3) managers with low risk aversion are attracted to high-power compensation packages. I develop a model where high-power incentive contracts act as screening devices, helping firms  attract less risk averse managers who will then make less distorted project selection decisions^ Optimal contracts trade off  the screening and effort-inducing Ill benefits  of  incentive contracts against the deviation from optimal risk sharing. The resulting equilibrium provides a new perspective on why some managerial contracts feature  such high-powered incentives, as well predictions for  the cross-sectional variation in the power of  incentive contracts. Table of  Contents Abstract ii Table of  Contents iv List of  Tables vi List of  Figures vii Acknowledgements viii Statement of  Co-Authorship. . . . . ix Chapter 1 Introduction 1 Chapter 2 Tort Liability and Capital Structure 6 2.1 Introduction 6 2.2 Motivation and Literature Review '. . • 9 2.2.1 Why is tort liability important, and to which firms?  9 2.2.2 Why tort liability matters from a capital structure perspective . 11 2.2.3 Tort liability and bankruptcy 14 2.2.4 Judgment proofing  16 2.2.5 Previous research 18 2.3 Basic Model 20 2.3.1 Continuous firm returns framework  20 2.3.2 Discrete firm returns framework  . 29 2.4 The Impact of  Liability Insurance . . 36 2.4.1 Discrete firm returns with insurance 36 2.5 Joint and Several Liability 41 2.5.1 Known returns case 42 2.5.2 Binomially distributed returns case 45 2.6 Two Period Model 51 2.6.1 Basic two period model 52 2.6.2 Two period model with tort liability '••••. 53 2.7 Conclusion .56 2.8 Bibliography '. 59 Chapter 3 The Capital Structure of  Insurers: Theory and Evidence 63 3.1 Introduction 63 3.2 The Optimal Capital Structure of  Insurers 66 3.2.1 Aggregate Uncertainty in Accident Losses 66 3.2.2 Uncertainty in Accident Probabilities 74 3.3 Evidence . . . 83 3.3.1 Introduction 83 3.3.2 Empirical Proxies and Estimation 84 3.3.3 Results 87 3.4 Conclusion 88 3.5 Bibliography 90 Chapter 4 Contracting With Agents of  Heterogeneous Risk Aversion 93 4.1 Introduction 93 4.1.1 Overview 93 4.1.2 Literature review 97 4.2 Model: Single Firm, Single Agent . . 101 4.2.1 First best solution 103 4.2.2 Hidden information:  the decisions of  risk averse managers 104 4.2.3 The optimal contract 105 4.2.4 Solution properties 107 4.2.5 Discussion . . 123 4.3 Model: Competitive Labour Market, Two Agent Types 124 4.3.1 Full information  case . ; 125 4.3.2 Private information  case ! . . . .128 4.4 Conclusion 142 3.5 Bibliography 144 Chapter 5 Conclusion 147 List of  Tables 3.1 Descriptive Statistics 84 3.2 Results ! 87 List of  Figures 2.1 Choice of  debt level given for  sizes of  judgment ( 35 2.2 Choice of  debt level under joint and several liability with known returns. 44 2.3 Choice of  debt level under joint and several liability with binomially distributed returns 49 2.4 Single large firm base case debt level 50 3.1 Optimal equity choice under different  degrees of  uncertainty in the probability of  loss . 78 Acknowledgements Vlll I would like to thank the faculty  members of  the Sauder School of  Business who helped me throughout the PhD program. In particular, I would like to thank Harjoat Bhamra, Gilles Chemla, Ruth Freedman, Ron Giammarino, Marcin Kacperczyk, Mo Levi, Alan Kraus, Cornelia Kullmann, Kai Li, and Tan Wang for  helpful  advice throughout my studies - some related to these essays, and some not. I would particularly like to thank Murray Carlson and Rob Heinkel for taking the time to serve on my thesis committee. I would also like to thank classmates past and present whose friendship  and great ideas were a source of  inspiration: Chris Bradley, Casey Clements, Julian Douglass, Shinsuke Kamoto, Caglar Kamu, Lars Kuehn, Kenny Pun, Thomas Ruf,  Jan Schneider, Daniel Smith, Issouf  Soumare, Bill Stewart, Richard Taylor, Marcus Xu and Longkai Zhao. I would like to make special mention of  my thesis advisor, Ralph Winter. There are no words sufficient  to describe the positive impact his influence  has had on me, academically, professionally  and personally, so I won't grasp for  them. It is not an exaggeration to say that this thesis would not have been completed had it not been for  his well-timed intervention. In the end it is impossible to name people who helped me complete this thesis without overlooking many more whose names could just as well be on this page. To those whom I have missed, I apologize. Finally, I would like to dedicate this thesis to my family,  whose interest in corporate finance,  is negligible, but whose interest in me finishing  this work has been great, and much appreciated. Statement of  Co-Authorship The second essay (Chapter 3) is written jointly with Ralph Winter and Dajiang Guo. My contributions to the paper included building and refining  the theoretical framework,  gathering the data, analyzing the data, and preparing the final manuscript. Chapter 1 Introduction In this thesis I examine three different  topics in corporate finance.  The exact nature of  the questions posed differ  across the three essays, but they are all questions of  importance to business entities. In the first  essay, I examine how firms  faced  with tort liability ought to adjust their capital structure, the mix between debt and equity securities. Tort liability has expanded enormously over the years. Firms must be aware that a substantial portion of  the firm's  assets and cash flows  are at risk of  being transferred  to tort claimants, should they win a legal judgment against the firm.  The law and economics literature has advanced a good deal of  study to ways in which firms  ought to seek to mitigate this problem. One possibility that has received scant notice from scholars working in the area is that firms  are able to alter their potential exposure to tort risk by making changes to their capital structure. Specifically,  corporations facing  potential legal risk may do better to finance  themselves with a greater proportion of  debt than they would otherwise. Most operational risks affect  the total value of  the firm's  cash flows  regardless of how claims to those cash flows  are structured. This is not the case with tort risk. When tort claimants win a judgment against a firm,  the amount they recover depends on what other promises have been made regarding how the firm's  assets will be distributed. In terms of  priority of  claims, in most jurisdictions, tort claimants collect ahead of  shareholders, but have equal or lower priority than unsecured creditors, and lower priority than secured creditors. This implies that when tort claimants win / • . . . a judgment against a firm,  they can collect fully  against equity holders. However, once the equity share has been exhausted and the firm forced  into bankruptcy, the tort claimants either share on a pro rata basis with unsecured creditors after  secured creditors have been paid fully,  or tort claimants make no further  collection whatsoever. I develop a simple model where a firm  trades off  the asset-shielding and tax advan-tages of  debt against the increased probability of  bankruptcy costs. I then consider extensions to the model, the first  considering the availability of  liability insurance, the second considering joint and several liability regimes. In the second essay, we explore the cross-sectional variation in insurers' capital structures: the choice by stock insurers- of  the mix of  equity and liability. As in the standard theory of  optimal capital structure in finance,  predictions of  the theory must rely on specific  capital market imperfections.  We focus  here on the simplest one: that issuing and maintaining additional equity is costly. Our model yields testable implications with a focus  (appropriate for  an analysis of  insurance markets) on the liability side of  the market. We develop the simplest model of  an insurance market with costly equity, in a two-period setting. For equity to have any role in an insurance market there must be aggregate uncertainty, or dependence among insured risks; the absence of  a law of large numbers means that equity is necessary to back up promises to pay claims in the event of  adverse realizations of  aggregate shocks. Accordingly, the key comparative static issue that we focus  on is the impact of  increasing aggregate uncertainty. We test the theory using cross-sectional data on U.S. property-liability insurers. The focus  is on tests of  two hypotheses. The first  is the implication of  the static model, that leverage is decreasing in aggregate uncertainty. The second is an impli-cation of  previous dynamic models of  competitive insurance markets that external equity is more costly than internal equity - specifically  that there is a positive cost to the "round-trip" of  distributing an amount of  cash then raising the same amount in external equity. We also offer  a link between the recent insurance market litera-ture and corresponding empirical results in tests of  capital structure for  non-financial corporations. In the third essay, I consider the problem firms  face  when contracting with man-agers when there is heterogeneity in risk aversion in the pool of  managerial labour. I motivate the essay with a number of  observations. The first  is that managers differ in their degree of  risk aversion, and that a manager's risk aversion is not observable. Second, higher managerial risk aversion is costly in two ways. First, higher risk aversion means that the manager puts a lower value on risky pay. This implies that the cost of  motivating effort  exertion is increasing in managerial risk aversion. However, managerial risk aversion is also costly in terms of  motivating correct project selection. When selecting projects, managers have an. incentive to choose those that best fit their own interests, as opposed to those of  firm shareholders. The greater difference in risk preferences  between risk averse managers and risk neutral shareholders, the greater will be the distortion imposed by managers selecting projects according to their own interests. A third observation is that the market for  managerial labour, like any labour market, is a competitive one. Firms compete with one another for  the services of preferred  managers, and managers will choose to work for  the firm that makes them the offer  they prefer. These three observations taken together imply the following.  From the second observation, it is clear that firms  prefer  lower risk aversion managers. From the third observation, they must compete against other firms  for  the services of  lower risk aversion managers. And from the first  observation, such competition is difficult, since a manager's risk aversion is his own private information.  Firms must therefore develop contracts which serve as screening devices, designed so that they will attract low risk aversion managers. Since all managers prefer  more pay to less, firms  cannot compete for  low risk aversion managers simply by raising wages. Since lower risk aversion managers put greater value on risky pay than high risk aversion managers, firms  have an incentive to offer  high-powered contracts as a screening device. Such contracts have greater appeal to the targeted low risk aversion managers. I develop a model where firms  must compete against one another in the managerial labour market to attract managers who are responsible for  both project selection and effort  exertion. In this setting, incentive contracts perform two functions.  The first  is to serve the traditional role of  motivating the correct effort  choice. The second is to act as a screening mechanism, helping firms  compete for  the services of  a lower risk aversion manager whose preferences  lead to better project selection. Chapter 2 Tort Liability and Capital Structure 2.1 Introduction Firms face  risk from a variety of  sources.. One type of  risk that has expanded enormously over the years is that posed by legal liability. Operating in an increasingly litigious society means that firms  must be aware that a substantial portion of  the firm's assets and cash flows  are at risk of  being transferred  to tort claimants, should they win a legal judgment against the firm.  The law and economics literature has advanced a good deal of  study to ways in which firms  ought to seek to mitigate this problem. The ideas range from working to avoid lawsuits in the first  place by exercising greater care, to purchasing insurance in order to substitute a sure loss for  exposure to the stochastic whims of  juries, to restructuring the firm so that there are fewer  assets exposed to legal liability. One possibility that has received scant notice from scholars working in the area is that firms  are able to alter their potential exposure to tort risk by making changes to their capital structure. Specifically,  corporations facing potential legal risk may do better to finance  themselves with a greater proportion of debt than they would otherwise. Since Modigliani and Miller first  posited that under a set of  restrictive assump-tions a firm's  capital structure does not matter, numerous models have emerged which attempt to demonstrate alternative circumstances under which a firm's  capital struc-ture might indeed impact the aggregate value of  securities issued by a firm.  To the extent that firms  have an optimal or "target" capital structure, it is most commonly modeled as a tradeoff  between some tax advantage provided by debt, versus some in-creased probability that the firm will be bankrupt and incur bankruptcy costs. This is the standard tradeoff  model. One way of  discussing the standard tradeoff  model is to consider the various parties' claims to firm cash flows.  The firm's  goal, when choosing its capital structure, is to maximize the value of  claims belonging to various groups of  security holders (usually, bondholders and shareholders, although more complex forms  are possible). Maximizing security holders' claims to assets entails minimizing the value of  claims that will accrue to other parties, such as the government .(taxes) and direct or indirect bankruptcy costs. The tradeoff  model has had little to say about the specific  type of  risk posed by a tort judgment. Most operational risks affect  the total value of  the firm's  cash flows  regardless of  how claims to those cash flows  are structured. This is not the case with tort risk. When tort claimants win a judgment against a firm,  the amount they recover depends on what other promises have been made regarding how the firm's  assets will be distributed. In terms of  priority of  claims, in most jurisdictions, tort claimants collect ahead of  shareholders, but have equal or lower priority than unsecured creditors, and lower priority than secured creditors. This implies that when tort claimants win a judgment against a firm,  they can collect fully  against equity holders. However, orice the equity share has been exhausted and the firm forced into bankruptcy, the tort claimants either share on a pro rata basis with unsecured creditors after  secured creditors have been paid fully,  or tort claimants make no further collection whatsoever. Thus the effect  of  tort risk on capital structure involves a tradeoff  of  its own. To the extent that tort liability adds risk to cash flows,  and decreases the expected value of  cash flows  that can be paid to other parties, an increase in tort liability increases the risk of  bankruptcy for  a given debt level. This may induce firms  to reduce debt. On the other hand, as firms'  increase the level of  debt in their capital structure, the more likely that the firm can take advantage of  tort claimants' relatively low priority to reduce the amount that they are expected to be paid. This countervailing effect sees firms  increasing debt as tort liability increases. The purpose of  this paper is to determine how firms  best ought to use debt, in the face  of  tort liability,'to maximize the aggregate value of  the firm's  securities. I develop a simple model where a firm trades off  the asset-shielding and tax advantages of  debt against the increased probability of  bankruptcy costs. I then consider extensions to the model. In the first,  the firm has the option to purchase liability insurance,in addi-tion to setting a debt level, to mitigate total expected judgment, tax and bankruptcy costs. In the second extension, I explore a situation where two firms  face  judgment risk in a joint and several liability regime. In this setting, each firm is liable for  one half  of  the total judgment, plus whatever portion the other firm is unable to pay. The section considers how the interaction between the two firms'  capital structure decisions affects  the equilibrium1 debt level. Finally, using a two-period version of  the model, I consider how firms'  optimal target debt level evolves as tort risk changes over time. 2.2 Motivation and Literature Review 2.2.1 Why is tort liability important, and to which firms? It is generally accepted that the number of  lawsuits in the United States has seen a remarkable increase over the years. Tillinghast Towers Perrin estimates that 2004 U.S. tort costs exceeded $260 billion, or 2.22% of  US GDP. That figure  represents an average per capita cost of  $886 for  every citizen of  the United States. Since 1950, the average annual percent increase in total tort costs has exceeded annual GDP growth by more than 2 percent. Commercial tort costs, the type most relevant to this paper, have grown at the fastest  rate of  late. Prom 1999 to 2004, commercial tort costs increased at an average annual rate of  11.6% per year. (Tillinghast Towers Perrin, 2006) An extremely litigious society, coupled with juries that over the years have awarded hefty  punitive damages with increasing enthusiasm, means that all economic actors are aware of  the substantial risk posed by the potential of  tort litigation. Not all firms  are equally likely to face  a lawsuit. Certain lines of  business natu-rally engender greater risk of  imposing harm on others, and suffering  a judgment as a result. Tobacco, waste management, firearms,  chemical manufacturing,  medical de-vices and pharmaceuticals are examples of  the industries where the very nature of  the business leads to risk of  imposing harm on others. This in turn leads to the potential for  litigation on a massive scale. States' 1990's litigation against the large tobacco companies, and a slew of  recent class action suits against Merck for  the manufacture of  Vioxx are just two examples. Firms affected  by asbestos litigation have been hit particularly hard. Over 6000 firms  have faced  asbestos-related lawsuits, and the vast majority of  these firms  were not involved in the manufacture  of  asbestos products. In more than 60 cases, the litigation led directly to the defendant  firm's  bankruptcy. (Carroll et al 2002) While tort risk varies across industries, no firm is immune, and operating in any line of  business can lead to litigation. The Loewen Group, an aggregator of  funeral homes, was involved in what appeared to be a minor dispute over a few million dollars in service contracts. This situation eventually led to a $500M judgment against Loewen, bankrupting the firm,  by the time a Mississippi jury was done deliberating the case. While firms  in certain industries are at particular risk of  finding  themselves defending  tort claims in court, the fate  can potentially befall  almost any firm. 2.2.2 Why tort liability matters from  a capital structure per-spective An argument that tort liability deserves special consideration when considering a firm's  capital structure must include an explanation of  how tort liability differs  from other forms  of  risk the firm faces.  If  tort liability were merely a stochastic reduction in the firm's  terminal cash flows,  analyzing it separately from other forms  of  cash flow risk would not yield any particular insight. The important distinction with respect to tort liability risk is that its potential impact on firm value depends on how claims to the firm's  cash flows  are structured. For example, unlike a $100M reduction in the value of  firm assets due to changing product market conditions, a $100M judgment against a firm does not necessarily reduce the value of  a firm's  assets by $100M. Tort claimants can collect their judgment only up to the value of  equity securities. Once the firm's  equity is exhausted and the firm is forced  into bankruptcy, tort claimants are left  to collect as much as their claim as they can from the firm's  assets after  more senior creditors have been paid.1 In the standard tradeoff  model, a firm faced  with a negative stochastic impact to cash flows  will tend to move to a lower debt level. . Expected tax savings are diminished, and the probability of  facing  bankruptcy is increased. This is not nec-essarily the case with tort liability. The dollar value of  firm assets actually paid to tort claimants is limited to the value of  the assets not promised to higher priority claimants - for  the purposes of  this paper, the debt holders. Thus while increased tort liability brings with it the increased probability of  bankruptcy, and therefore  the increased probability of  incurring bankruptcy costs, it also brings greater potential savings due debt. In addition to debt providing a tax advantage, for  a firm faced with tort liability it also provides an "asset shielding" benefit:  a dollar of  cash flows promised to debt holders cannot be fully  expropriated to pay tort claimants.2 1See Painter (1984) for  a detailed description of  tort claimant priority with respect to other creditors. Depending on the jurisdiction and other circumstances, tort claimants have either (a) lower priority than secured creditors and equal priority to unsecured creditors, or (b) lower priority than all debtholders. For the purposes of  this paper, tort creditors are assumed to have lower priority than debtholders. "Debt" as described in this paper should therefore  be interpreted as being 'an instrument that gives its holder higher priority than tort claimants. 2 The comparison between the standard tradeoff  model and a model involving tort risk starts in Section 3 with the simplest comparative static results. The fundamental  difference  between an operational risk that may potentially neg-atively impact cash flows  and tort liability is that - unlike tort risk- the potential loss from the operational risk cannot be mitigated by adjusting the firm!s  relative amounts of  debt and equity financing:  While optimal capital structure is affected by the operational risk, the operational risk to cash flows  is not affected  by capital structure. On the other hand, the magnitude of  potential tort liability is affected  by capital structure. When considering how to deal with tort risk, the firm doesn't only consider the costs that would come with financial  distress; financial  distress provides the advantage of  shielding some of  the firm's  cash flows  from tort claimants' reach, reducing the ex ante value of  their claim. Because these two effects  work in opposite directions, the direction of  the impact of  tort risk on optimal capital structure is not immediately obvious. In other ways expected tort liability is similar to a firm's  expected tax liability. Both depend on the firm's  capital structure, and represent expected claims on cash flows  to be paid to parties other than a firm's  security holders. However, a firm's tax liability is not in itself  stochastic. While the exact realization of  a firm's  tax bill is uncertain ex ante, it is a deterministic function,  of  the firm's  eventual cash flows, promised payments to debt holders, and the residual cash flows  accruing to equity holders. A firm's  expected payment to tort claimants also depends on the firm's  cash flow  and the relative mix of  debt and equity. Tort liability is not a deterministic function,  in that the future  decisions of  judges and juries are uncertain. Another crucial difference  is that tax liability does not tend to push a firm into bankruptcy,-while an unfavourable  tort judgment most certainly can.3 Finally, as I demonstrate in Section 5, tort risk involves interesting interactions or externalities among firms  in their capital structure decisions. This moves capital structure from the realm of  a single agent decision to game theory. The externalities give rise to multiple equilibria, where aggregate debt levels can end up being much higher or much lower than they would be in the absence of  the interaction between agents' decisions. 2.2.3 Tort liability and bankruptcy To date, there is little in the way of  research into how firms  adjust capital structure when faced  with tort liability. However, there are numerous cases, many high profile, where tort judgments have pushed firms  into bankruptcy. This possibility must be taken into account when firms  determine their capital structure. Any assumption to the contrary strains credulity. Among the most high profile  instances of  a firm going bankrupt as a result of lawsuits is the Johns-Manville company. One of  the earliest cases of  a "mass tort", the asbestos manufacturer  soon became deluged by lawsuits from plaintiffs  alleging health problems as a result of  exposure to the firm's  product. In 1982, Manville filed 3Of  course, it is possible that an unpaid tax bill on a firm's  past earnings could lead to the firm being forced  into bankruptcy. However, this situation would be analogous to the tax collector winning a "judgment" against the firm,  and would therefore  fit  into the model as a form of  tort risk liability. . for  bankruptcy under Chapter 11, "not because of  an inability to meet its current debts, but rather because of  its anticipation of  massive asbestos personal injury claim liability in the future"  (Vairo 2003) While Manville was one of  the first  firms  to go bankrupt as a result- of  asbestos liability claims, it was certainly not the last. According to the Rand Institute for Civil Justice, over 6000 firms  in nearly every industry have faced  at least one lawsuit related to asbestos liability. For most firms  the cost is negligible relative to firm assets, but over 60 firms  have filed  for  bankruptcy as a direct result of  asbestos litigation. (Carroll et al 2002) And cases where litigation leads to bankruptcy are not limited to asbestos. Other high profile  cases include that of  Dow Corning, which filed  for  bankruptcy in 1992, awash in litigation stemming from injuries caused by breast implants, and A.H. Robins, which filed  for  bankruptcy in 1985 as a result of  litigation related to its Dalkon Shield intrauterine device. There is little in the way of  comprehensive evi-dence linking tort liability and bankruptcy. However, a wealth of  anecdotal evidence suggests that this type of  problem does occur frequently,  and to large firms,  and thus motivates an examination of  the logic of  optimal capital structure for  firms  facing tort risk. 2.2.4 Judgment proofing Using capital structure to reduce exposure to tort liability risk is only one means by which firms  are able to reduce their exposure to lawsuits. Other methods exist, with one of  the most common, and most commonly studied, is to create "narrow entities". That is, to the extent that certain risky lines of  business likely to lead to tort liability can be isolated from the rest of  the firm,  then this is what the firm should do.4 If  the risky behaviour gives rise to a lawsuit, then plaintiffs  will be left to sue an entity whose pockets are much less deep than those of  the firm as originally constituted. Ringleb and Wiggins find  evidence related to this form of  judgment proofing. Their proxy for  lawsuit liability is industry worker exposure to carcinogens. After controlling for  various other possible explanatory factors,  they find  that the proportion of  small firms  operating in a given industry is significantly  positively correlated with the degree of  worker carcinogen exposure in that industry. They take this as evidence that firms  operating in industries associated with potential judgment risk tend to be smaller, and therefore  have fewer  assets. The propensity of  firms  in risky industries to operate as narrowly as possible does not obviate the need to consider asset shielding through capital structure as an alternative, or in some cases complementary, technique. In some instances, it may not be possible to separate risky activities from less risky activities. While it was 4See Lopucki (1998) and Roe (1986) for  discussion of  these ideas in detail. logical for  the RJ Reynolds tobacco business to be split from the Nabisco division, it would be impossible for  RJ Reynolds to further  separate sales of  the cigarettes that cause cancer from those that don't. When a risky activity is the core of  the firm, further  separation is simply not feasible. Further, in some instances, a cost must be incurred to set up narrow entities. If  a risky activity is integral to a firm's  broader activities, and the efficiencies  from keeping the risky division internally outweigh the foregone  expected tort judgment savings from not spinning it out, the firm will keep the division internally. There may be legal impediments to judgment proofing.  For example, legislators may mandate that firms  performing  certain activities have sufficient  resources to pay potential litigants in the event that they cause a tort. This requirement would normally be satisfied  either by minimum asset requirements or by compulsory liaibility insurance.5 As well, U.S. law provides a means of  reducing the advantage of  judgment proof-ing through the creation of  a narrow entity. Courts have the power to "pierce the corporate veil"; that is, in some instances courts hold shareholders of  a tortfeasor firm liable beyond the value of  their shares. This is most likely to occur exactly when the firm has structured itself  narrowly. According to Bergmann, "corporations are expected to operate with a certain minimal level of  assets that takes into account 5See Shavell (2005) for  a more detailed discussion of  these types of  solutions to judgment-proofing problems, as well as an analysis of  how such requirements affect  incentives with respect to making care decisions to avoid accidents in the first  place. the particular nature and risks of  that enterprise". (Bergmann 2004) In other words, creating a narrow entity solely for  the purpose of  performing  risky tasks in order to shield assets against simply may not work. Finally, even in cases where as narrow an entity as possible is established, and assuming that the entity has been established in such a way that the courts will not engage in veil piercing, that entity will still face  potential tort judgment liability. That firm must make a capital structure decision in the presence of  that liability, making the research questions posed by this paper relevant for  that firm. 2.2.5 Previous research In the law and economics literature, there has been some research that considers the role bankruptcy (and by extension, capital structure) has to play in the context of a firm that faces  tort liability. However, the focus  has largely been on how tortfeasors behave given the potential for  insolvency. Huberman et al (1983) consider how an economic entity will make an insurance decision when liability has the potential to make it insolvent. They find  that bankruptcy protection leads firms  to lower levels of insurance than would be optimal otherwise.6 Kornhauser and Revesz (1990) consider how the potential for  insolvency will affect  a firm's  decision as to the level of  care it will take to avoid incurring a lawsuit, under different  liability regimes. These, and 6The result is driven by the notion that an insurer must pay for  damages caused by the firm even in states where the firm  is insolvent. As such, when insurance is fairly  priced from the insurer's perspective, the firm  pays for  coverage even in bankruptcy states where it has no need for  insurance. Because the firm is paying "too much", it has an incentive to back away from full  insurance. other similar papers, generally takes the probability that the firm becomes insolvent to be exogenous. That is, the firm does not make a capital structure decision in these models. As I make clear in this paper, this is problematic, as capital structure is an endogenous decision made in the context of  all risks facing  the firm,  including tort risk. Other papers consider how different  regulations regarding lender liability affect firms'  actions. Heyes (1996) studies how making lenders liable for  some part of  the damages caused by their debtors affects  both firms'  cost of  capital and level of  care taken to avoid causing torts. Pitchford  (1995) considers a similar question. Both conclude that the equilibrium cost of  capital will (most likely) increase, but that the effect  on firms'  decision with respect to level of  care is ambiguous. Yahya (1988) is closest in spirit to this paper, in that he allows firms  to choose both a level of  debt and a degree of  care, and considers how the firm's  decision changes under a variety of  liability regimes. By contrast, this paper seeks to make no recommendation as to how the legal system ought to be structured. Rather, the question posed here is to consider how the tort liability system, as constituted, will cause firms  to respond to tort liability with changes in their capital structure. 2.3 Basic Model 2.3.1 Continuous firm  returns framework Consider a firm with the opportunity to pursue a one-period investment project. For simplicity, assume that the risk free  interest rate is zero, and that the project's risk is entirely idiosyncratic. Investors are fully  rational and risk neutral.7 Static tradeoff  between cost of  financial  distress and tax savings The firm must choose its time zero capital structure, which will be a combination of  equity, with a time zero market value So, and one period debt with a promised time 1 payment D, which has a time zero market value B0. Define  V 0 as the sum of the time zero market value of  the securities issued, V 0 = S0 + B0. The project's terminal value is stochastic, and has a cumulative distribution func-tion G(V)  and associated probability density function  g(V),  with <2(14^) = 0 and G(Vjnax) = 1. The firm faces  a tax rate r on the time 1 payoff  to equity , while debt holders' returns are not taxed. In the event that the realization of  the project's terminal value is less than the face  value of  the debt, i.e. V  < D, then the firm is bankrupt, and incurs fixed  bankruptcy cost C. By assumption, C < V nym. 7This is equivalent to assuming that the risk facing  the firm is idiosyncratic, and investors are^  well diversified. The market value of  debt is given B0= J  (V  - C)g(V)d(V)  + J  Dg(V)dV  (2.1) V m i „ D while the market value of  equity is given Vmax So= J (V-D)(l-  r)g(V)dV  (2.2) D The expected bankruptcy cost, Co, is the cost of  bankruptcy should it occur times the probability that the firm goes bankrupt. This can be expressed D Co — J Cg(V)dV  (2.3) r^nin while the expected tax bill, To, is v m a x m To = y \V  - D)rg(V)dV D The expected value of  the firm's  cash flows  is Knax E(V)  = J Vg(V)dV m^in = Bo + So + Co + To (2.5) in contrast to the market value of  the firm's  securities, V 0, which is Vo = i?o + So = E(V)-Co-To  (2.6)-The firm's  capital structure does not have an effect  on the probability distribution governing the total cash flows  to be shared between claimants. Therefore,  the value maximizing level of  D is that which minimizes the sum of  expected bankruptcy costs and the expected tax bill. Since ^jj > 0 and ^ < 0, there is a value D*, V min < D* < V max that maximizes ex ante firm value Vo- This promised debt level occurs where ^ -DD 3D • Introduction of  legal liability Consider now the same firm,  faced  with the probability p that a tort litigant will appear, • successfully  sue the firm,  and win a judgment J to be paid from the terminal asset value V. The claim has higher priority than equity, but lower priority than debt.8 The expected payout to tort claimants (and the expected cost of  tort liability), Jo, is given / V M AX  . D+J  \ Jo=P\ J J ^ V ) d V + / (V-D-  C)g(v)dv\  (2.7) WJ  D+C / The whole expression is multiplied by p, which is the probability that the plaintiffs  win a judgment against the firm.  The first  term inside the brackets represents the range of  terminal asset values where the firm is solvent, and must pay the tort claimants in full.  The second term represents terminal asset values where the firm is bankrupt, and the tort claimants only collect their judgment after  bankruptcy costs are paid and debtholders are paid fully.  For asset values below D + C, tort claimants receive 8This is a simplification,  in that in some jurisdictions other priority rules may apply. nothing. The market value of  debt is D vm a x D+C B0= J (V  - C)g(V)dV  + J Dg(V)dV  + p J (V  - D - C)g{V)dV  (2.8) V MI N  D D The first  two terms represent the market value of  debt if  there were no tort risk. The third term represents the impact the expected tort judgment has on the value of debt.9 The market value of  equity is expressed m^ax So = J (V-D)(l-r)g(V)dV D ( D+J  V MA X  \ J (V  - D)(l  - r)g(V)dV  + J J{\  - r)g{v)dv\  (2.9) D D+J,  / The first  term is a standard expression for  the after-tax  value of  equity. The sec-ond term represents the expected cost to equity holders if  the firm loses the tort judgment.10 Expected bankruptcy costs are D D+J C0 = J Cg(V)dV  +p J Cg(V)dV  (2.10) V MI N  D ' 9Since debtholders have priority, the size of  the judgment, J, does not affect  the value of  debt. The effect  is through the increased probability of  bankruptcy in the event of  losing the judgment. When tort claimants win the case, debtholders bear some portion of  the bankruptcy costs for  terminal asset values between D and D + C. 10Note'that the expected transfer  to tort claimants is also calculated on an after-tax  basis. The first  term is the expected bankruptcy cost in the absence of  tort liability, where the second term measures the expected increase in bankruptcy costs brought about / by tort liability. Finally, the expected tax bill is • Vmax Vmax T0 = (1  -p) j (V-D)rg(V)dV  + p J (V  - D - J)T 9(V)dV D D+J T So (2.11) (1  ~r) Once again, the market value of  the securities the firm issues depends on the amount of  debt issued: V0 = So + Bo = E{V)-Co-T 0-J 0 (2.12) The goal of  the firm is to set the debt level that maximizes the aggregate value of  debt and equity. As can be seen from equation 2.12 , this is equivalent to setting the debt level, D*, that minimizes the sum of  expected bankruptcy, tax and tort judgment costs. The reformulated  first  order condition is therefore  that the optimal debt level is chosen such that 8 ( c ° + g + J o ) = 0. More detailed analysis depends on the distribution of  firm asset returns. Com-parative statics are unwieldly in the general case. As such, further  analysis of  how the optimal debt level changes is best conducted by studying specific  distributional forms  for  the firm's  asset returns. Uniform  distribution Comparative statics are facilitated  by making a distributional assumption regard-ing firm's  asset returns. Assume that V  is distributed uniformly  between V m\n and Knax, i.e. that a(V)  — 77—K7—.  The expected costs are then: ' m a x ^ m i n C[D+pJ-V min} 6 0 - — ^ — r y : — ' max v mm ^ = p i e > - « c + D + iJ-v„ ) ( 2 1 4 ) 'max m^in rp \D2 + PDJ  + \vJ 2 ~(D+  PJ)V max + "max m^m Taking the derivative of  the sum of  the cost functions,  setting to zero and solving for D yields D* = V m a x - - + ^—^pJ  (2.16) r r This can be compared to the firm's  optimal debt level in the absence of  tort liability, which is D*J =0 = VUx - - (2-17) T  • Since the tax rate, r, is defined  over 0 < r < 1, the optimal debt level is increasing in the both the size of  the judgment to be paid if  the firm loses the case (J) and the probability of  having to pay the judgment (p).  So for  the case where the firm's cash flows  are distributed uniformly,  an increase in the expected judgment pj leads to an increase in the optimal face  value.of  debt. This implies that the expected costs stemming from the increased probability of  bankruptcy brought about by an increase in expected tort liability are outweighed by the asset shielding advantages of a relatively high debt level. Importance of  claim priority The move away toward debt in the face  of  an increasing expected tort judgment highlights the importance of  the priority of  claims. In the above model, tort claimants collect only after  debt holders have been paid. Tort claimants are, in effect,  similar to involuntary subordinated debt holders.11 To see the importance the asset shielding effects  of  debt, consider the solution when tort claimants have priority. Here the firm still chooses a debt level to minimize the sum of  Jo, C 0 and To; to reflect  the change in priority, Jo = pJ.  Firm returns are assumed to be uniformly  distributed over VJnin, V m a x . For this specification,  the optimal debt level is D* = Knax - — - pJ  (2.18) r With the asset shielding benefits  of  debt gone, the firm reduces its target debt level as the expected judgment increases. This specification  allows to consider a simple decomposition of  the two effects  on the optimal debt level that stem from an increase in judgment liability: the increase 11This analogy is only approximate. J  represents the face  value of  the subordinated debtholders' claim, while, in a debt issue, it would be most likely that p = 1, as most debt issues require an attempt at repayment in all circumstances (lottery bonds and catastrophe bonds being exotic exceptions). in expected bankruptcy costs, and the debt advantages of  asset shielding. Since D* is the optimal debt level in the presence of  increased bankruptcy costs without the benefits  of  asset shielding, the bankruptcy cost effect  can be defined  as D* - DJ =q = -pJ  (2.19) The shift  caused by the asset-shielding advantages of  debt, when debt has higher priority, is then calculated D* -D* = -pJ  (2.20) T In the uniform distribution case, with 0 < r < 1, the asset shielding advantages of debt outweigh the associated bankruptcy costs, and increases in tort liability lead to an increase in the optimal debt level. Fraudulent conveyance An important consideration for  firms  that choose to use capital structure as a defence  against tort liability is whether or not their chosen capital structure will stand up to tort creditors' efforts  to collect. As is true with any judgment proofing technique, there exists the risk that capital structure defences  may be overturned. In this eventuality, the court would rule that setting a high debt level was done solely for  the purpose of  reducing the claim or tort creditors. The court would then be in a position to declare that the firm's  capital structure amounts to a form of  fraudulent conveyance, and award tort claimants higher than anticipated priority, thus rendering the firm's  efforts  to insulate its security holders from tort risk moot. In the one-period model described above, the risk that the firm's  defensive  strategy would be overturned can be introduced relatively simply by assuming that whatever the firm's  chosen capital structure, there exists the probability o; that tort claimants will be awarded higher priority than debtholders. In this case, for  a firm whose asset returns are uniformly  distributed (as above), the firm's  problem becomes to minimize the sum of C[D  + pJ-V min} v max v mm rr • \D2 + PDJ  + \pJ 2 - ( D + PWnax  + V ^ f 0  9 9 . To  = T  ; _ (2.22) 'max ymin , J,  (• iCP-AO  + D + jJ-V:.ax ) • Jo  = apJ  + p(l-a) A — —— : (2.23) "max •'min The optimum debt level is D* = y m a x - - + l~T~ apJ  (2.24) T T which implies that debt is decreasing in the probability a that the firm's  defences will be overturned. This is not surprising, given that the expected asset shielding benefits  of  debt decrease in a, while the expected tax and bankruptcy costs remain the same.12 As the expected judgment cost pJ  increases, changes in the optimum debt level 12The increase in the expected judgment cost comes at the expense of  debtholders, the value of whose claims would fall  should a court award tort claimants higher priority. Since debt is fairly priced, increases in a decrease the time 0 value of  debt. . are no longer necessarily strictly increasing. It is readily apparent that dD* 1 - r - a (2.25) d(pJ)  r implying that if  the sum of  the tax rate, r, and the probability of  claim priority being changed in favour  of  tort creditors, a, is greater than 1, then increases in the expected judgment result in the firm using less debt. The intuition is that for  sufficiently  high values of  a, the expected asset shielding benefits  of  debt are reduced to the point that they are overtaken by the associated increased expected bankruptcy costs. 2.3.2 Discrete firm  returns framework An alternative specification  is one where the firm's  returns follow  a discrete prob-ability distribution. Consider the firm in the previous section. Instead of  firm cash flows  following  a continuous distribution between V M- M  and V MA X,  suppose that the cash flows  follow  a binomial distribution. At time 1, the firm's  return is Vl  with probability (1 — q) and VJJ  with probability q. All other variables are as defined  in the previous section. Tradeoff  between financial  distress and tax savings I begin by reviewing the standard static tradeoff  model. The market value of  debt and equity, as well as expected bankruptcy costs and the expected tax bill, depend Bn = So = on the promised debt payment D. The market value of  debt is > D for  D < V L (1 - q)(V L — C) + qD for  V L < D < V H (1 - q)V L + qV H-C  for  V H  < D while the market value of  equity is ((l-q)V L + qV H-D)(l-r)  for  D <V L q(V H-D)(l-r)  for  V L < D <V H 0 for  V H  < D The  expected tax bill is To  — So 1 — T while expected bankruptcy costs are 0 for  D < V L (1 - q)C for  V L<D<V H C for  V H  <D The expected value of  the firm's  cash flows  is . Co ={ (2.26) (2.27) (2.28) (2.29) E(V{)  = (1  -QH)V L + QV„ • — BQ + SQ + T 0 + CO (2.30) while the market value of  the firm's  securities is given Vo = -Bo + So = E{VT)-T 0-CQ (2.31) This implies that the market value of  the firm is maximized when the term (T0 + C 0 ) is minimized. It is clear that for  D <VH,  §§• < 0. However, since bankruptcy costs are fixed  should they occur, and the probability of  incurring these costs only increases at the debt levels D = {Vr,,  VJI  }, the solution will be one of  these two values: The marginal expected tax savings from moving from debt level D — VL  to D — VH  are T o y L - T o y ^ r q i V H - V i ) .(2.32) while the marginal expected bankruptcy costs are (1 -Q)C (2.33) Define  C as: C = ~ V L) • (2.34) Then for  C > C the firm will set D = V L, and for  C < C the firm will set D = V H. Introduction of  legal liability Consider now the same firm,  faced  with the probability p that a tort litigant will appear, successfully  sue the firm,  and win a judgment J  to be paid from the terminal asset value V,  with priority higher than equity, but lower than debt holders. As in the previous section, the firm's  problem is to set a debt level that maximizes the market value of  its securities. However, unlike the previous section, the firm must additionally consider the risk posed by tort liability. On one hand, tort liability increases the risk of  bankruptcy for  a given level of  debt, which would suggest a shift to a lower promised debt payment. On the other, where debt holders have priority over tort claimants, higher debt means a lower expected payment to tort claimants, suggesting a shift  toward a higher promised debt payment. The net effect  will depend on parameter values. Once again, it is possible to consider a finite  number of  debt levels. For all debt levels D < VH-,  a small increase of  £ in D decreases the expected tax payment. However, if  at debt level D, shifting  to D + e does not lead to an increase in the expected bankruptcy cost, then D is not a potential solution. An £ increase in debt always leads to a decrease in the tax bill, and sometimes to a decrease in the expected judgment cost. Using this logic, one can easily show that the set of  admissible debt levels which may solve the firm's  problem are D — {VL  — J,VL,VH  — J,  VJJ}• Given the possible debt levels, and assuming J  < (V H  — V L) and VL  > C + J, the expected cost of  tort liability is pj for  D = V L - J JO  = pqj for  D = V L,V H-J > (2.35) 0 for  D = VI H \ Expected bankruptcy costs are 0 for  D = V L-J p(l  - q)C for  D — Vl (2.36) Co — > (1 - q)C for  D = V H~J (1  -q + pq)C iorD = V H \ B 0 = The market value of  debt is V L-J  for  D = V L - J V L-p{l-q)C  for  D = Vl (1 - q)(V L -C) + q(V H  - J)  for  D = V H  - J (1 - q)(V L -C) + q(V H-pC)  for  D = V H and the market value of  equity is (1 - r)(q(V H  - V L) + (1 - p)J)  for  D = V L - J (1  - t-)q(V H  - V L - pJ)  for  D = V L (1 - r)q(  1 - p)J  for  D = V H  - J 0 for  D = V H  • Tax is once again defined  relative to equity, Sn = (2.37) (2.38) % = 1-T So (2.39) The solution to the firm's  problem is to choose a promised debt payment from the set D = {Vl — J, V ,^ Vjj — J, Vjj} such that the sum of  expected bankruptcy, tax and judgment costs are minimized. Define  C(D)  as the total expected costs from choosing debt level D. The total expected costs from each of  the four  choices are C(V L - J) = qT(V H-V L) + [p  + r(l-p)}J (2.40) C(V L) = p(l-q)C  + pq(l-r)J  + qT{V H-V L) (2.41) C(V H  - J) = (1  - g)C + q\p + t(1  - p)]J (2.42) • C{V„) = [(1  -q) + qp]C (2.43) As the firm moves progressively through to higher debt levels, the bankruptcy costs increase. To offset  this effect,  the expected cost of  the tort judgment decreases (in bankruptcy states the firm doesn't pay), and expected tax costs decrease as well. Depending on the parameter values, any of  the four  debt levels may prove to be optimal. Unlike the continuous case with a uniform distribution, the optimal debt level is not necessarily increasing in J.,  At low levels of  J, firms  are more likely to choose to accommodate the probability of  facing  a judgment by choosing either Vl  — J or Vh  — J]  as the potential cost of  the judgment, J, increases, at some point the firm will no longer choose to accommodate the judgment, and will shift  to either debt level Vl  or Vh.  If  the potential judgment becomes sufficiently  large relative to bankruptcy costs and other parameters, the firm will choose the maximum debt level, Vh-The following  diagram illustrates the potential for  some firms  to go through the entire range of  possible debt choices depending on the level of  the potential judgment, J.  For the set of  {V L = 100, V H  = 200,p = 0.5, q = 0.5, r = 0.5, C = 65}, the optimal debt level D* is on the vertical axis with the. potential judgment, J, plotted on the horizontal axis: 2 0 0 1 8 0 1 6 0 14 0 12 0 10 0 Figure 2.1: Choice of  debt level for  given sizes of  judgment In this example, in the absence of  tort liability, the firm chooses Vl-  As the potential judgment, J, increases, the firm initially chooses Vl  — J,  meaning that debt decreases dollar for  dollar as J  increases. At a critical point, the optimum jumps to VH  — J  - As the potential judgment continues to increase, the firm eventually switches to a debt level of  Vl-  In this range, the optimum debt level is locally insensitive to changes in J.  As J  becomes sufficiently  high, the firm moves to maximum debt, VH-The non-monotonicity arises in the discrete case because at low levels, small -in- • creases in the potential judgment do not warrant the increased risk of  bankruptcy. However, as the size of  the potential judgment rises relative to potential bankruptcy costs, eventually the asset-shielding benefits  of  debt outweigh the costs from bankruptcy, and the firm chooses to increase debt. The discrete case highlights the importance of  assumptions regarding asset returns. For different  probability distributions of  firm returns, the optimal capital structure response to changes in tort liability will differ. 2.4 The Impact of  Liability Insurance In some circumstances, firms  may have the opportunity to buy liability insurance. Tillinghast Towers Perrin estimates that in 2003, over $91 billion in tort costs were covered by firms'  insurance policies. When liability insurance is an option the firm must make a joint capital structure-insurance coverage decision in order to maximize firm value. This section considers this decision problem. f 2.4.1 Discrete firm  returns with insurance Assume that the structure of  operating cash flows  and tort liability is the same as in the previous discrete returns case. Now, the firm may choose to buy insurance, up to the value of  the judgment, J, which pays off  in the event that the firm loses a lawsuit and must pay a tort judgment. Assuming that the insurance is fairly  priced, I  dollars of  coverage costs pi. Proposition 1.1 If  a firm  whose returns are binomially distributed chooses to insure, it will do so fully,, Proof:  The advantage of  insurance is that it can be used to eliminate the prob-ability that a judgment against the firm will cause it to incur bankruptcy costs. The disadvantage is that even under fairly  priced insurance the premium is greater than the expected judgment cost, as long as the firm chooses a debt level such that there is some positive probability of  bankruptcy. In the discrete case, buying anything less than full  insurance does not decrease the probability of  bankruptcy, while every dollar of  coverage purchased does reduce the asset shielding advantage of  debt. Therefore, if  it is advantageous to buy the first  dollar of  insurance, it is more advantageous still to buy J dollars of  insurance.13 QED • In a setting with insurance, the firm has the same capital structure options as in the previous section, as well two new choices. The set is D — {VI  — J,  VL,I=J,  VL,VH~ J jVh >i=j,Vh},  with Vlj=j  and Vhj=j  representing choices of  debt level where the firm has chosen to fully  insure against judgment liability.14 The various costs of  each choice are / \ pJ  for  D = V L - J JO  = pqj for  D = V L,V H-J  ? ( 2-44) 0 for  D = VLJ =J,  VH,I=J,  VH Cn = 0 for  D = V L-J,V LJ =J p{  1 -Q)C for  D = V L (1  -Q)C for  D = VJ-J,VH,I=J (1  -q + pq)C for  D — Vh (2.45) 1 3 See Huberman et al (1983). 14Assume that Vij=j  implies a promised payment to debtholders of  Vi  — pJ,  and a promised payment to the insurer of  pJ.  Further assume that the insurer has priority, and Vi  — C > pJ, guaranteeing that the insurer will be paid. TN  — < r(q(V H  - V L) + (1 - p)J)  for  D = V L - J rq(V H  - V L) for  D = V LJ= J rq{V H-V L-pJ)  for  D = V L rq(l  — p)J  for  D = V H-J 0 for  D = V HJ= J,V H as well as the insurance premium: (2.46) /n = pj for  D = VL,I=J,  VH,I=J 0 otherwise (2.47) Proposition 1.2 When insurance is available, VL,I=J  dominates VL  — J-Proof:  Because a firm choosing VL  — J  never faces  bankruptcy, it does not take advantage of  the asset shielding effects  of  debt. For this firm,  the cost of  buying in-surance is equal to the expected payment to judgment holders if  uninsured. However, insurance allows the firm to take on a higher debt level, D = VL,  providing expected tax savings of  TJ,  without incurring bankruptcy risk. QED Define  the expected total costs for  a given debt level D and insurance choice I  as C(D,I).  For the various combinations to be considerd, the expected total costs are C.(V L,J) = pJ  + rq(V H-V L) (2.48) C(V L, 0) = p(l-q)C  + pq(l-T)J  + rq(V H-V L) (2.49) C(VH-J,  0) = (1 -q)C + q[p  + r(l-p)}J (2.50) C(V H,J) = (1 -q)C + pJ (2.51) C(VH,  0) = [(1 -q)+qp]C (2.52) Depending on the parameter values, any of  the five  choices can be optimal. Of particular interest is the choice of  {Vj/,</}, where the firm chooses the high debt level but also purchases insurance. Firms making this decision are the only ones that "overpay" for  insurance, to the extent that they surrender the asset shielding advantage of  debt and pay for  coverage in states where judgment holders would have been unable to collect. Despite this, it can still be an optimal decision if  the tax savings brought about by being able to choose the high debt level Vh,  without fear of  increased bankruptcy risk brought about by tort liability, are sufficient. However, firms  will only ever consider one of  {D , 1} — {VH  — J,  0} and {VH,  J}-Note that firms  are indifferent  between the two choices where P r ^ ' (2.53) (1 -p) (1 -q) The left  side is the likelihood ratio of  losing the tort judgment, and the right side is the likelihood ratio of  realizing the high return multiplied by the tax rate. When the left  side is greater than the right, implying relatively higher probability of  losing the lawsuit, the firm will consider {Vh  — J, 0}. When making this choice, the firm avoids overpaying the insurance premium, but accepts increased taxes when it realizes high returns and does not lose the lawsuit. When the right is greater the firm instead considers {VH,  J}-  Here, the firm enjoys maximum tax savings, but at the cost of paying for  insurance it does need when it realizes low terminal asset values. Note that which set the firm considers is independent of  both J  and C, meaning that firms  will only ever consider one or the other.15 Therefore  there is no set of  parameters {p,  q, r } where changes in J  and C can produce as many as five  different  optimal debt choices. In general, the availability of  insurance reduces firms'  propensity to accommodate a potential judgment by choosing either D = Vj, — J  or D — VH  — J',  the former  is never chosen, and the latter considered only when the right side of  equation 2.53 is greater than the left.  This implies the existence of  a greater number of  states where the firm chooses a higher level of  debt. However, there also exist parameter values for  which firms,  who in the absence of insurance would have chosen V L or VH,  shift  to {14, J}  or {VH,  J}-  This suggests that while the presence of  insurance leads to more debt, it also leads to a greater number of  states where tort creditors recover fully.  Insurance leads to more firms  with deep pockets.16 To the extent that tort judgments are legitimate attempts to redress those who have been harmed in some way, this is a socially desirable effect. 15Of  course, how the cost of  either {VH  — J,  0} or {VH,  J}  compare to the costs of  the other three options depends critically on the relative values of  C and J. 1 6 Strictly speaking, the deep pockets belong to the insurance companies with whom the firm has contracted. Prom the plaintiff's  perspective, this distinction is not important. 2.5 Joint and Several Liability An essential extension of  the analysis is to consider how a firm's  behaviour changes when it is dependent on the outcome for  other firms.  This situation arises where firms r are jointly and severally liable for  a given tort. In the simplest example of  how this type of  liability works, a plaintiff  sues two defendants  who both contributed to causing her harm. If  a judgment is found  in the plaintiff's  favour,  each defendant  is ordered to pay half  the judgment. However, if  one of  the defendants  becomes insolvent, the other becomes responsible for  whatever remaining portion of  the judgment needs to be paid. This extension is far  from being an esoteric detail. Joint and several liability is now standard for  many types of  torts in many juriscitions. According to the 2004 report by Tillinghast Towers Perrin, "there appears to be a shift  in the types of  liabilities that make up the total tort costs in the U.S., from individuals suing individual entities to groups of  plaintiffs  taking legal action against one or more entities". In this circumstance, capital structure choice is the outcome of  a game. The capital structure choices of  a set of  firms  who share liability for  a given tort become interdependent. The externalities among firms  give rise to the possibility of  multiple equilibria. The implication is that firms'  decisions with respect to capital structure now depend on the decisions of  other firms:  specifically,  those of  their co-defendants. As is demonstrated below, circumstances can emerge where a firm would choose a relatively conservative debt level, if  it knew that its potential co-defendants  would do the same, thereby committing to being solvent to able pay their share of  the potential judgment. However, if  the co-defendants  choose higher debt levels, implying that the firm would be left  on its own to cover the cost of  the entire judgment, the firm's decision would change; it too would shift  to a higher debt level. 2.5.1 Known returns case First, consider a case with two firms,  each of  whose asset value will be V  at the end of  the period. At that point, the firms  will lose a tort case with probability p, in which case they will be jointly and severally liable for  paying the judgment J. If both firms  are solvent, they each owe | to the tort claimants. Should one firm not be able to pay the judgment, then the other is responsible for  the full  amount. The firms  each choose a debt level, Dt. It is straigtforward  to show that each firm's  optimal debt level will always be one of  the three values {V  — J, V  — V} . 1 7 Without loss of  generality, I restrict the analysis to these three values. The optimal level for  each firm depends not only on the parameter values for {p, r, C, J}, but also on the other firm's  choice of  debt level. This occurs because each firm must consider whether or not the other firm has chosen a capital structure 17The tax and asset shielding advantages to debt financing  are continually increasing in D, while the expected bankruptcy costs only jump at specific  debt levels. The critical levels at which an epsilon increase in debt will (sometimes) increase the probability of  bankruptcy are V  — J  and V  — V  is the highest possible debt level, as there are no asset shielding or tax benefits  to choosing a debt level beyond this point. that will leave it solvent and able to pay its share of  the judgment should the firms lose the lawsuit. In terms of  impact on the other firm,  the choices V  — J  and V  — j can be grouped together, since both of  these levels leave the firm able to meet its share of  the obligation. However, choosing a debt level of  V  imposes an externality on the other firm;  should the case be lost, the other firm will be faced  with a bill for the full  judgment. Expected costs from each choice of  debt level must be calculated based on the other firm's  decision. A cost function  Ci(Di, Dj) is defined  as the combined expected bankruptcy, tax and tort judgment costs for  firm  i, given that firm  i chooses debt level Di and firm  j choose debt level Dj. The cost functions  to be considered are: Ci(V-J t{V-J iV-^})  = ^]p + T+{l-p)T] CI(V-J,V)  = JIP+(1-P)T} Ci{V-±{V-J,V- J-})  = J-  b + (1 - p)T] Ci{V-lv)  = ±\p+(l-p)T]+pC Q(V,{V-J,V-^V})=PC Looking at the cost functions,  it is immediately apparent that Di — V  — ^ is never a best response when the other firm sets Dj = V; it is dominated by a symmetrical response of  Di = V.  It is also clear that Dt — V  — J  in response to Dj — {V  — J, V  — is dominated by D% = V  - f :  Therefore,  {D u D3] = {V  - J, V  - J},  {V  - J,  V  -— are not possible equilibria. ! The remaining possible equilibria are {D i: Dj} — {V  — — {V,  V},  and {  V  — J,  V}.  Which equilibrium' will prevail depend on the size of  the possible judgment J  in relation to the other variables. When J  <C P P + (1  ~p)T {D i;Dj} — {V  — j;, V  — is the only possible equilibrium. When J  > 2C- P p+ (1 -p)r {Di,  Dj} = {V,V}  is the only possibility. However, in the region C- P < J  <2C- P p+(l-p)r~ ~ p+(l-p)r either of  the two equilibria is possible. While it is not difficult  to show that the firms would prefer  the {Di,Dj}  = {V  — — equilibrium, the firms  do not necessarily have the opportunity to choose. Once one firm has adopted the high capital structure, the other must follow  suit, and neither will have an incentive to deviate. A simple diagram illustrates this point, for  parameter values {V  — 10,p = .5, r = .5, C = 3}: Figure 2.2: Joint and several, liability and known returns Values of  J  are plotted on the bottom axis, while debt levels are on the vertical axis. The two curves represent optimal responses, depending on the other firm's choice. The curve which is initially more steeply sloped represents optimal choices of  Di when the other firm's  debt level is Dj = V.  The curve that is initially less steeply sloped represents optimal choices of  Di when the other firm's  debt level is Dj — {V  — J,  V  — In the region J  < 2, the optimal debt level for  both firms  is V  — When J  > 4, both firms  choose V.  However, for  2 < J < 4, two equilibria are possible. Either {D u Dj} = {V  - - {},  or {A, Dj} = {V,  V}. 2.5.2 Binomially distributed returns case Consider , two firms,  identical to those described previously. Instead of  facing a certain return V, each firm faces  symmetric,independently binomially distributed returns. That is, each either returns Vl  or Vjj  at time 1, and each has the same probability of  realizing a high return, ^ = qj = q. The firms  will be ordered to pay a judgment J  with probability p. Again, the firms  are jointly and severally liable. The firms  each choose a debt level, {Di,  Dj}. Again, each firm need only consider a finite  number of  potential debt levels. The initial set to be considered is D — {V L — J,V L — ^,V L,V H  — J,VH  — VII}-  The complication comes when each firm must consider the expected bankruptcy, tax and tort judgment costs associated with each debt level in response to the possible choices of  the other firm.  An equilibrium is a situation where each firm's  choice is a best response to the other firm's  debt level, recognizing that each firm's  asset returns are stochastic. The expected costs associated with firm Vs decision to choose debt level Di, given that firm  j chooses Dv are defined  as Ci(Di,  Dj). The cost functions  are as follows: Ct{V L-J,{V L-J,V L-J-}) = qT(V H-V L) + t{p  + T + {l-p)T)  (2.54) Ci(V L-J,{V L,V H-J,V H-^}) qr (V H  + ^ (p(2 + q)(  1 - r) + r) (2.55) Q(V L-J,V H) .= qr{V H-V L) + J(p+(l-p)T)  ' - (2.56) Ci(V L-l{V LlV H-J,V H-l}) qr(V H  -V L)+p(  1 - qfC  + J-pr{-  + - - 1 - (1 - q)q) (2.57) I  T  p qr {V H  - V L) + J  (|(1 - r) + r ) (2.58) C,(V L-3-,VH) qr(V H  - V L) + p(l - q)C;+ J-  (p(l - r) + r ( l - pq)) (2.59) = qr(V H-V L)+p(l-q)C  + ^pq(l-T) Q(V l,{V l,V h-J,V h-^}) qr(V H  - V L) + p( 1 - q)C + J-pq{  1 - r)(2 -C',(V,: V/,) = qr(V H-V L)+p(l-q)C  + (l-r)pqJ Ci(V H-  J,{V L~ J,V l-^}) = (l-q)C  + ^(pq(l-r)  + 2rq) Ci(V„  - J,  {V L, V H  — J,V H  — = (1 - q)C + J- (pq(  1 -r)(2-q)  + 2rq) ' QXVh-^VH) • = .(l-q)C  + Jq(p  + T(l-p))  . = (l-q)C  + ^q(p  + r(l-p))  . = {1  - q)(l  + pq)C + Jq(p  + r{l  - p)) Ci(v H-lvH) (l-(l-p)q)C  + Jq(p  + r(l-p)) (2.68) CI(VN,  DJ) (1-(1  -p)q)C (2.69) While this set of  cost functions  is difficult  to analyze analytically, it is possible to do some numerical experimentation. For several sets of  parameter values, it becomes clear that while all of  the debt levels may be optimal in some circumstances, it is generally true that equilibria involve firms  choosing to match each other's debt level; the relevant cost functions  to consider are then defined  by equations 2.54, 2.58, 2.61, 2.64, 2.67 and 2.69.18 . Further, it is apparent that both firms  choosing the debt level Vl  — J  will not be an equilibrium. Provided that both firms  choose Vl  — J,  each has an incentive to move to Vl  — -f  • Since neither set of  choices ever results in either firm going bankrupt, the firms  prefer  Vl  — , as the higher debt level provides a lower expected tax bill, with expected bankruptcy costs and judgment costs remaining unchanged. The optimal decision depends on the parameter values p, q, C, VL,  VH,  T  and J. Comparisons are probably most relevant when made as follows.  . 18It may be possible that two firms  will choose different  debt levels, with the higher debt firm's choice imposing a greater externality on the other firm.  However, for  this to be an equilibrium, .it would have to be the case that Ci(H,H)  — Ci(H,L)  > Ci(L,H)  — Ci(L,L).  This condition can be interpreted as being that a shift  to the higher debt level has a greater negative impact on the other firm when the other firm is already at the higher debt level. There is not any evidence that a set of parameters meeting this condition, and being otherwise consistent with the model, exists. For the sake of  exposition, I examine the capital structure decision of  two identical firms,  to be held jointly and severally liable for  the amount J  should they lose the court case, with the parameters {Vl  = 50, V H  = 100, p = . 5 = .5, r = .5, C; = 10} held constant, as the aggregate amount of  the potential judgment J  varies. I compare this with the debt decision taken by one firm,  faced  with the same potential liability J, with parameters {V L = 100, V^ = 200, p = ,h,q = .5,r = .5, C = 20}. This is relevant because aggregate "industry" revenues and bankruptcy costs are the same as for  the two smaller firms,  as is the potential judgment. As J  varies (values on the horizontal axis), the two firms'  choice of  debt level is plotted: 1 0 0 r 9 9 r 9 8 r 9 1 9 6 9 5 9 4 Figure 2.3: Joint and several liability and binomially distributed returns In the range 0 < J  < 6.67, the equilibrium is A = D0 — V H  - J.  For J  > 7.27 the equilibrium is the maximum possible debt level, Di — Dj = Vh-  For the range 6.67 < J  < 7.27, either of  the other two equilbria are possible; the firms  either both select VH  — J,  or they both select VH-The decision of  the larger single firm,  faced  with the entire liability itself,  is plotted: 1 0 Figure 2.4: Single large firm base case debt level Here, the firm initially chooses debt level VH  — J,  and shifts  to VH  when J  = 13.33. In the example, it's apparent that the effect  of  joint and several liability on the capital structure decision is ambiguous. At low levels of  J, the two firms  choose to accommodate not only their own initial share of  the judgment, but also that of the other firm,  recognizing that their co-defendant  could go bankrupt. For values of J < 6.67, tort claimants' expected recovery is higher than the one-firm case.19 For J  > 7.27, both firms  will shift  to the highest possible debt level, V}/, and tort claimants' expected recovery drops to zero. In the single firm case, this shift  does not occur until J  > 13.33. For values of  J  such that 6.67 < J  < 7.27, two equilbria are possible. The firms  will either both accommodate the full  share of  the judgment by setting DI  = DJ  = VH  — J,  or both firms  will shift  to the highest possible debt level VH-Generally, joint and several liability serves to reduce debt levels and increase 19In the one firm case, the single firm has realize the high asset return for  the tort claimants to be able to collect the judgment should they win. The probability of  their having a claim against a solvent defendant,  conditional on having won the case, is q. On the other hand, when faced  with two defendants  each choosing a debt level of  VH  — J,  only one need be solvent. The probability of being able to collect is q(2  — q), which is greater than q. expected tort claimant recovery at low judgment levels, while it decreases expected recovery at higher debt levels. When the cost of  losing a judgment is low relative to the costs associated with going bankrupt, both firms  choose a capital structure which would allow them to pay should they realize high returns. Tort claimants end up benefitting  from a "diversification"  effect.  Rather than being exposed to the risk that a single defendant's  deep pockets will be emptied by the vagaries of  business risks, defendants  have two entities to pursue, and enjoy the increased probability that at least one's pockets will remain deep. As the potential size of  the judgment increases, however, the defendants  start to impose externalities on each other. To protect themselves from having to pay the other's share of  the judgment, both choose aggressive capital structures to insulate themselves against the potential judgment. Aggregate debt therefore  tends to increase, and tort claimants recover in fewer  states of  the world than they would against a single, larger defendant. 2.6 Two Period Model An extension which adds some richness to the analysis is to consider how firms will make capital structure decisions as tort liability evolves over time. The antici-pated risk of  losing a major lawsuit is not static. As new information  emerges about the likelihood that the firm has caused a tort against another party, or about the magnitude of  the harm caused, all market participants will reasonably update their expectations about the probability of  the firm having to pay tort claimants. By the same token, neither is capital structure static. Firms have the flexibility to increase or decrease their debt level as time goes on, continually trading off  the asset and tax shielding benefits  of  debt versus the expected cost of  financial  distress. By considering a two-period model, it is possible to consider how a firm's  capital structure changes through in time, in response to changes in tort risk. 2.6.1 Basic two period model Consider a firm pursuing a project with a two period life.  At the end of  the first period, the firm receives an update as to the distribution governing the project's final distribution. At the end of  the second period, the terminal asset value is realized. The distribution of  the time 1 reported asset value is governed by probability density function  g(Vi)  and cumulative distribution function  G(\i),  with G(V\  r n i n ) = 0 and G(V lmax) = 1. The realization of  Vi,  which is the signal received at time 1, is the time 1 expectation of  the eventual time 2 cash flow  realization. This time 2 cash flow realization, V 2, is distributed with probability density function  g(V 2) and cumulative distribution function  G(V 2), with G(V 2mm) = 0 and G(V 2in;iX)  = 1. V 2min and V2max are defined  such that Ei(V 2 \ Vl) = V,. This implies that £0(Ki) = E0(V 2).20 The firm finances  its operations through a mix of  one-period debt and equity. At 20An example of  the type of  situation this set of  distributions is meant to describe would be as follows.  A firm receives an updated signal, Vi, uniformly  distributed between 50 and 150. The firm's eventual value, V2, will be uniformly  distributed between V\  — 50 and Vi + 50. As such, Ei(V2)  = Vi, while the time zero expectations are Eo(Vi) = £0(^2) = 100. time zero, the firm issues debt with a face  value of  Di, payable at time 1. At time 1, after  receiving the updated signal Vi, the firm chooses a face  value of  debt for  the second period, D2, payable at time 2. Taxes are payable at both time 1 and time 2, as a fraction  of  asset value at that date. Debt payments shield assets from the tax collector, so the taxes payable at a given date t are T t = r(V t — Dt) whenever V t > Dt. zero otherwise. Finally, bankruptcy costs C are incurred at either date whenever V t < Dt. This can be interpreted as costly renegotiation. By assumption costs, there are no costs associated with adjusting capital structure. In the absence of  any frictions,  the firm chooses a debt level at date t — 1 such that minimizes Et_i[C t + T t]. Because the relative.distributions at both dates between Vtmm and V t m a x are the same, the firm chooses the same relative debt level each period, denoted Dt*.21 2.6.2 Two period model with tort liability Consider now a firm faced  with tort risk, such that there is some chance that they will pay a judgment J  at time 2. The time zero risk of  having to pay the judgment-is At time 1, the firm receives an updated signal about the lawsuit's prospects. 2 1 Of  course, the absolute second period debt level will be higher when VI  > EQ(VI),  and lower when V 1 < E0(VI). With probability \ the suit is found  to have no merit (lawsuit risk falls  to zero), and with probability \ the plaintiffs'  chances of  winning the suit improve to p. By assumption, the firm's  time 1 tax charge does not change, and is still defined  as T\  = R{VI  — DI).  However, the.risk of  financial'  distress increases, as security holders take the time 2 judgment risk into account when valuing their claims. The time zero expectation of  the time 1 bankruptcy cost becomes £O(CI)  = C The firm solves for  the promised time zero debt payment which minimizes EQ[C\  + Ti]. Because tort risk increases the probability bankruptcy, and period 1 debt shields assets from the tax collector only, the optimum debt level D\ is lower than the optimum level in the case where expected tort liability is zero at time 1. At time 1, if  the risk of  tort liability disappears, the optimum time 2 promised debt payment is D*2. the same as the optimal level in the case without tort, liability. On the other hand, if  the probability of  having to pay a judgment increases to p,.the firm chooses D\ so as to minimize the sum of  E\ [C2  + T 2 + J2}  • This is essentially the same problem as defined  in the single period model described earlier in this paper. Whether or not DZ,  is a higher relative debt level than D* depends on distributional assumptions about the distribution of  V..  As shown in the one period model, however, if  Vi is uniformly  distributed, then the debt level is increasing in expected tort liability. 1 rDi 1 j-D1+PE^{J) - / giVjdV  + - g^dV ^ V i m i n J V l m i n Since this is the case where expected tort liability increases from time zero to time 1, for  a uniform distribution it will be the case that D\- • \ This leads to an interesting conclusion. For at least some distributional assump-tions about firm returns, the further  resolution of  uncertainty about tort risk leads to an increase in the firm's  relative debt level, whether that resolution increases or decreases tort risk. While this is somewhat counterintuitive, it can be explained as follows.  At time zero, the probability of  having to pay a judgment is p/2. For a given time 1 face  value of  debt, this risk increases the expected bankruptcy cost, providing an incentive to lower debt. However, because the judgment, if  eventually paid, will only be paid at time 2, the promised time 1 debt payment does not provide any asset shielding benefits. At time 1, there are two possible resolutions of  uncertainty about tort risk. In the case where tort liability disappears, the probability of  bankruptcy for  a given relative debt decreases. Since the tax shielding benefits  of  debt don't change, the new optimum relative debt level is higher than it was at time 0. In the case where tort liability increases, the expected bankruptcy costs also increase. However, promising a debt payment at time 2 helps shield the firm's  security holders from tort creditors, which is not true of  the promised time 1 debt payment. For distributions where the asset shielding benefits  of  debt outweigh the increased expected bankruptcy costs, even increased tort liability at time 1 will lead to the firm choosing to increase its debt level. 2.7 Conclusion This paper seeks to explain how tort liability will affect  a firm's  optimal cap-ital structure. While other papers have made the point that limited liability will affect  economic agents' incentives with respect to tort risk, very few have sought to endogenize the firm's  decision about in which states it will be solvent. A key characteristic of  tort risk is that its impact on cash flows  available to security holders depends on the structure of  security holders' claims. Put another way, capital structure matters greatly when determining the potential expense payable to tort claimants. . Recognizing this, firms  with exposure to tort liability will have an incentive to adjust capital structure to respond optimally. The lower creditor priority of  tort claimants implies two effects  when debt and tort risk interact. The first  is that tort liability brings about an increased probability of  bankruptcy. Where this effect predominates, the firm will choose to move away from debt. The second effect  is that debt provides an asset shielding advantage, preserving cash flow  rights for  the firm's  debt holders at the expense of  tort claimants. Where this effect  is dominant, increased tort risk will cause the firm to choose more debt. I specify  two simple models to examine the interaction of  these effects,  one where firm returns are distributed continuously over an interval, and another where firm returns are distributed binomially. The different  results from these two illustrations demonstrate the importance of  assumptions regarding firm cash flows.  Depending on the nature of  the firm's  returns, and the values of  the various input parameters, either the bankruptcy effect  or asset shielding effect  can dominate. I also consider how liability insurance affects  the outcome. Fairly priced liability insurance is in effect  overpriced for  any firm with positive probability of  bankruptcy, due to the asset shielding effects  of  debt. However, the model in this paper demon-strates that there are circumstances where firms  will still choose to purchase insurance. The model also indicates that the availability of  insurance can lead to greater amounts of  debt being issued, at the same time as providing tort litigants with deeper-pocketed targets. I test how firms'  capital structure decisions change when several smaller firms  are jointly and severally liable for  a judgment, and compare their behaviour to that of  a larger entity faced  with the same potential judgment. I find  that for  relatively low tort amounts, debt levels tend to be lower, and tort claimants' expected recovery greater. However, for  higher judgment amounts, debt levels tend to increase, and tort claimants will expect to recover less. Finally, I examine how the capital structure decision changes as liability evolves through time. In the model, I find  that the resolution of  uncertainty about tort risk leads to an increase in the debt level, whether the resolution is one of  lower tort risk or higher tort risk. Tort liability is a major source of  risk for  firms  today. I have explained why it is unique, and why firms  must consider its unique properties when determining the optimal capital structure. Empirical work studying how firms  do adjust their capital structure to address changes in tort risk is a potentially fruitful  avenue for  future research. 2.8 Bibliography [1] Avila-Nores, Matias and Stephen L. Schwarcz, "Ring-Fencing and Other Bankruptcy-Remote Techniques", Duke University School of  Law Working Pa-per, 2002. [2] Bergmann, Karyn S., "Bankruptcy, Limited Liability and CERCLA: Closing the Loophole and Parting the Veil", University of  Maryland School of  Law Working Paper Series No. 2004-02, 2004. [3] Carroll, Stephen J., Deborah Hensler, Allan Abrahamse, Jennifer  Gross, Michelle White, Scott Ashwood and Elizabeth Sloss, "Asbestos Litigation Costs and Com-pensation", RAND Institute for  Civil Justice Documented Briefing,  2002. [4] Dahiya, Sandeep and David Yermack, "Wealth Creation and Destruction from Brooke Group's Tobacco Litigation Strategy", New York University Law School Working Paper No. CLB-00-007, 1999. [5] Delaney, Kevin J., "Power, Intercorporate Networks, and 'Strategic Bankruptcy'", Law & Society Review, _ 1989, Vol. 23, No. 4, 643-666. [6] Fleming, John G., "Mass Torts", The  American Journal  of  Comparative Law, Summer 1994, Vol. 42, No. 3, 507-529. [7] Hansmann, Henry and Reinier Kraakman, "Toward Unlimited Shareholder Lia-bility for  Corporate Torts", The  Yale  Law Journal,  May 1991, Vol. 100, No. 7, 1879-1934. [8] Heyes, Anthony G., "Lender Penalty for  Environmental Damage and the Equi-librium Cost of  Capital", Economica, May 1996, Vol. 63, No. 250, 311-323. [9] Huberman, Gur, David Mayers and Clifford  W. Smith, Jr., "Optimal Insurance Policy Indemnity Schedules", The  Bell Journal  of  Economics, Autumn 1983, Vol. 14, No. 2, 415-426. [10] Kornhauser, Lewis A. and Richard L. Revesz, "Apportioning Damages Among Potentially Insolvent Actors", The  Journal  of  Legal Studies, June 1990, Vol. 19 No. 2, The Law and Economics of  Risk, 617-651. [11] Kraus, Alan and Robert H. Litzenberger, "A State-Preference  Model of  Optimal Financial Leverage", The  Journal  of  Finance,  September 1973, Vol. 28, No. 4, 911-922. [12] Lopucki, Lynn M., "Virtual Judgment Proofing:  A Rejoinder", The  Yale  Law Journal,  March 1998, Vol. 107, No. 5, 1413-1434. [13] Malani, Anup and Charles Mullin, "The Effect  of  Joint and Several Liability on the Bankruptcy Rate of  Defendants:  Evidence from Asbestos Litigation", University of  Virginia Law School Working Paper, 2004. [14] Painter, Christopher M.E., "Tort Creditor Priority in the Secured Credit System: Asbestos Times, the Worst of  Times", Stanford  Law Review, April 1984, Vol. 36, No. 4, 1045-1085. [15] Pitchford,  Rohan, "How Liable Should a Lender Be? The Case of  Judgment-Proof  Firms and Environmental Risk", The  American Economic Review, Decem-ber 1995, Vol. 85, No. 5, 1171-1186. [16] Priest, George L., "Understanding the Liability Crisis", Proceedings  of  the Academy of  Political Science, Vol.  37, No. 1, 196-211. [17] Roe, Mark J., "Corporate Strategic Reaction to Mass Tort", Virginia  Law Re-view, February 1986, Vol. 72, No. 1, 1-59. [18] Ringleb, A1 H. and Steven N. Wiggins, "Liability and Large-Scale, Long-Term Hazards", The  Journal  of  Political Economy, June 1990, Vol. 98, No. 3, 574-595. [19] Schwartz, Alan, "Products Liability,. Corporate Structure, and Bankruptcy: Toxic Substances and the Remote Risk Relationship", The  Journal  of  Legal Stud-ies, December 1985, Vol. 14, No. 3, Critical Issues in Tort Law Reform:  A Search for  Principles, 689-736. [20] Shavell, Steven, "The Judgment Proof  Problem", International  Review of  Law and Economics, 1986, Vol. 6, 45-58. [21] Shavell, Steven, "Minimum asset requirements and compulsory liability insur-ance as solutions to the judgment-proof  problem", RAND  Journal  of  Economics, . Spring 2005, Vol. 36, No. 1, 63-77. [22] Swiss Re, "Sigma: The economics of  liability losses - insuring a moving target", 2004, No. 6. [23] Tillinghast - Towers Perrin, "U.S. Tort Costs: 2004 Update". [24] Tillinghast - Towers Perrin, "U.S. Tort Costs and Cross-Border Perspectives: 2005 Update", 2006. [25] Tung, Frederick, "Taking Future Claims Seriously: Future Claims and Successor Liability in Bankruptcy", Working Paper, 1998. [26] Vairo, Georgene, "Mass Torts Bankruptcies: The Who, The Why and The How", Loyola Law School Research Paper No. 2003-21, 2003. [27] White, James J., "Corporate Judgment Proofing:  A Response to Lynn LoPucki's The Death of  Liability", The  Yale  Law Journal,  March 1998, Vol. 107, No. 5, 1363-1412. [28] Yahya, Mom A., "Bankruptcy and Torts", Working Paper, 1998. Chapter 3 The Capital Structure of  Insurers: Theory and Evidence 3.1 Introduction In the simplest economic models of  insurance markets, which ignore transactions costs of  any kind, risks are priced at actuarially fair  values. This prediction depends on one of  two sets of  assumptions: the pooling theory of  insurance assumes that in-sured risks are independently distributed and large in number; the transfer  theory of insurance assumes that risks are independent, of  aggregate wealth in the economy and can be transferred  through the issuance of  equity to a perfect,  capital market (Mar-shall (1976)). Recent research in insurance economics has shown that the observed dynamics of  insurance premiums and contracts can be explained only by a failure  of both sets of  assumptions. Aggregate uncertainty, combined with imperfections  in the equity market, can disrupt the transfer  of  risks to the capital market in ways that explain insurance market dynamics (e.g., Gr0n (1994), Winter (1988,1994)). This connection is not surprising, since imperfections  of  some sort are necessary to explain even the existence of  insurance intermediaries. The empirical tests in this recent literature have focussed  on time series implications of  insurance pricing and capital flows. . This paper explores the cross-sectional variation in insurers' capital structures: the choice by stock insurers of  the mix of  equity and liability.1 As in the standard theory of  optimal capital structure in finance,  predictions of  the theory must rely on specific  capital market imperfections.  We focus  here on the simplest one: that issuing and maintaining additional equity is costly. Our model yields testable implications with a focus  (appropriate for  an analysis of  insurance markets) on the liability side of  the market. Section 2 of  this paper develops the simplest model of  an insurance market with costly equity, in a two-period setting. For equity to have any role in an insurance ^he capital structure decision for  insurers, being a financial  intermediary, is different  from the decision faced  by non-financial  firms.  Non-financial  firms  have some underlying assets which generate cash flows;  the capital structure decision relates to how to finance  those assets by apportioning claims to cash flows  between debt and equity holders. Insurance companies' liabilities are the insurance policies themselves, whice arise naturally in the course of  doing business. The question of  how much equity to maintain relates to what kind of "cushion" the firm  requires to credibly back the policies it issues. This problem is similar to the one made by banks; faced  with a given level of  deposits (liabilities), banks must determine how much equity it requires to maintain capital adequacy. market there must be aggregate uncertainty, or dependence among insured risks; the absence of  a law of  large numbers means that equity is necessary to back up promises to pay claims in the event of  adverse realizations of  aggregate shocks.2 Accordingly, the key comparative static issue that we focus  on is the impact of  increasing aggregate uncertainty. We consider separately the cases of  aggregate uncertainty in the loss incurred conditional upon an accident and uncertainty in the probability of  an accident (i.e. dependence among the events of  individual accidents). In the former  case, the total equity issued by a competitive insurance market is increasing in the degree of uncertainty (and linear in a parameterized example). In the latter case, equity may be increasing then decreasing as a function  of  uncertainty. In both cases, the ratio of equity to revenue is increasing in uncertainty. Section 3 tests the theory using cross-sectional data on U.S. property-liability insurers. While the theory is developed for  competitive markets, by assuming that each insurer is operating in a different  set of  one or more competitive markets, we can use firm-level  data in the tests. The focus  is on tests of  two hypotheses. The first  is the implication of  the static models that leverage is decreasing in aggregate uncertainty. The second is an implication of  previous dynamic models of  competitive insurance markets (Gr0n (1994) and Winter (1994)) that external equity is more costly than internal equity - specifically  that there is a positive cost to the "round-aggregate uncertainty is necessary, that is, in the limit as the number of  consumers gets large. With independence, the law of  large numbers would allow the risk of  bankruptcy to be avoided by a vanishingly small amount of  equity per policy. trip" of  distributing an amount of  cash then raising the same amount in external equity. Previous tests of  this implication focus  on the time series behavior of  insurance premiums. The empirical analysis here is complementary, based not on prices but directly on capital structure decisions. The paper also offers  a link between the recent insurance market literature and corresponding empirical results in tests of  capital structure for  non-financial  corporations: Titman and Wessels (1988) and Rajan and Zingales (1994) find  negative relationships between leverage and past profitability;  an explicit dynamic theory and tests are offered  by Fischer, Heinkel and Zechner (1989). 3.2 The Optimal Capital Structure of  Insurers We describe the capital structure choice of  an insurance firm in the simplest pos-sible model. The key assumption must be that risks are dependent, i.e. subject to aggregate uncertainty or common factors.  We consider separately the cases of dependence in the events of  accidents and dependence in the size of  losses incurred. 3.2.1 Aggregate Uncertainty in Accident Losses t Assumptions We consider a competitive market for  insurance. On the demand side of  the market, a large number of  individuals each face  with a known probability p the loss of  wealth. The size of  the loss is itself  random, taking on the value H  with probability A and L with probability (1 — A). If  the risks faced  by individuals were independently distributed then - given a large number of  individuals - insurance would be provided. at a fair  premium with no need for  equity. The optimal capital structure would (in. the limit) have zero equity. We introduce a role for  equity by assuming that the random losses faced  are dependent among individuals. In fact,  for  simplicity, losses are identical for  those experiencing an accident. In short, each individual faces  a two-stage lottery, with "accident - no accident" in the first  stage and "L or H"  in the second. Across individuals the first  stage outcomes are independently distributed, while the second-stage outcomes are identical. The individuals are expected utility maximizers and the gain from exchange in the insurance market arises because they are risk averse. We take the simple case of identical individuals, with initial wealth W  and utility U,  where U'  > 0 and U"  < 0. Ex. ante, a large number of  stock insurers issue equity and then issue insurance policies. An insurance policy is assumed to be non-participating. That is, the contract with any individual specifies  a payment that is contingent only on the individual's loss experience: II  dollars if  the individual experiences a loss of  L, I H  dollars with a loss of  II.  The premium is denoted by P.  We constrain the insurance contracts and equity to satisfy  a limited liability constraint, so that the contracts promised by the insurer must be credibly backed by the equity issued. We denote by E the equity per policy issued. A second constraint is that the promised payment in any accident state cannot exceed the accident loss in that state. This can be justified  by a moral hazard assumption that an individual has the ability to cause an accident intentionally. In a perfect  capital market, the cost of  issuing and maintaining equity would be zero. Equityholders would be indifferent  between investing through the insurance corporation and investing through their personal portfolios.  It is evident that in reality equity cannot be issued by an insurer and maintained without limit at zero cost. The costs include agency costs of  having corporate management intermediate between investment in assets and shareholders; the administrative costs of  issuing equity; the signalling costs of  issuing equity and the double-taxation of  corporate income.3 We do not model these costs explicitly, but simply assume that equity cannot be raised at zero cost. Specifically,  we assume that it costs (1  + c)E dollars to raise E dollars of  equity, which is returned to claimants on the firm's  assets. The term cE represents the net cost of  maintaining equity. Equityholders price equity according to the expected value of  net payments that they are to receive; this reflects  an assumption that the uncertainty in losses, while not diversifiable  in the insurance market, is diversifiable  in the stock market. Interest rates are zero. The supply of  insurance is taken to be competitive, which means that any capital structure E and policy (P, II,  IH)  consistent with zero expected return to equityholders will be supplied if  it is demanded. On the demand side, the individuals 3It is evident that means of  distributing cash to shareholders other than by dividends cannot be relied upon as costless alternatives. choose the most preferred  policy among policies offered  by the market. This model yields, as an equilibrium, the choice of  an insurance policy that max-imizes the expected utility of  the individual among all the policies yielding zero ex-pected return to stockholders. The issue of  concern is how the equilibrium values of. equity and the structure of  liabilities vary with uncertainty in losses. Remarks This is the simplest model within which we can address the impact of  dependence in risks and costly equity capital structure decisions. Several features  of  the simple model abstract from reality. First, we have taken the form,  of  the insurance contract, the nonparticipating contract, as exogenous. This can be justified  formally  with an assumption that an individual can verify  only his own accident experience. It includes the simplification  that no mutual insurance is available. Second, in this static model we do not capture any distinction between the costs of  maintaining equity, and the costs of  adjusting equity. The evidence from the recent literature is that this distinction is important for  explaining the dynamics of  pricing and capital flows.  In the empirical section, we shall in fact  offer  some evidence of  the cost advantage of internal capital - and, implicitly, of  the value of  extending this model to a dynamic context. Equilibrium Consider first  the payoffs  to equityholders and individual demanders of  insurance, under the contract [E,  P, I L, I H]  when this contract is offered  to all individuals.4 The payoffs  to an individual who does not experience an accident is W  — P.  The payoff to an individual who experiences an accident with loss X,  for  X  = L or H  is W  — P — X  + Ix  • The net payoff  to equityholders (per policy issued) in the event that the common accident loss is X  is — cE + P — pX,  since a proportion p of  individuals experience an accident. The contract offered  in a competitive insurance market will maximize expected utility subject to three constraints. The first  is a limited liability constraint, that the payment to accident victims in each event X  must not exceed the sum of  internal equity, (1 — c)E + P.  That is, plx < (1 — c)E + P.  The second is a participation constraint for  insurers, that the expected profit  be non-negative: — cE + P — P{XIH  + (1 — \)IL] > 0. The third is the constraint that Ix < X . The following  results are easily proved. Proposition 3.1: If  c — 0, then the equilibrium insurance policy involves full coverage of  each loss. Lemma 3.1: With  c > 0, the participation constraint is binding and: (a) the constraint Ii  < L is binding: IL  — L. 4It is convenient to consider the equity E as one component of  the contract; it backs the promise to pay the claims II  and IH• (b) the limited liability constraint is binding in the event H  : pin = (1 — c)E + P (c ) I h < H Proposition 1 is the standard perfect  capital market benchmark. Lemma 1 is for the case of  c > 0. Here, without the "moral hazard" constraint that II < L, low losses would actually be more than fully  covered.5 The lemma allows us to simplify the contract specification  and payoffs:  A contract can without loss of  generality now be described as a pair (P, E). Individuals receive a net payoff  of  W  — P  in any event except a high-loss accident, and in the event of  a high-loss accident they receive W-P-H+(P  +: (1 - c)E)/p = W  — H  + (^)P  + ^E . The gross payout to shareholders is zero in the event of  high accident losses (where the limited liability constraint is binding), so that the expected profit  to shareholders from issuing a contract (P,  E) is -cE + P-p[\I H  + (l-X)I L} Using Lemma 1 (a) and (b), this expected profit  can be written - ( c + A(1 - c))E + (1 - X)P  - p(l  - \)L In sum, we can characterize the equilibrium insurance contract as the solution to the following  problem: max (l-p\)U(W  - P)+p\U  (w  - H+  + (3.1) 5 This result follows  because the events of  an accident are independent across individuals, and therefore  the market offers  wealth transfers  between the events of  "accident" and "no accident" at an actuarially fair  rate. The individual optimum therefore  requires the equality of  marginal utility in the event of  no-accident and expected marginal utility conditional upon an accident. To achieve this equality, since high losses are not fully  covered, low losses must be more than fully  covered. subject to — (c + A(1 — c))E + (1 — X)P  — p(l  — X)L  — 0 (3.2) Letting the multiplier on the constraint be /./,. the first  order conditions with respect to E and P  respectively are: A(1 — c)U'(-)  — (c + A(1 — c))(i  = 0 (3.3) — (1 — p\)U'(W  — P)  + A(1 — p)U'(-)  + (1 — X)p  = 0 (3.4) where [/'(•) = U'  (w  - H  + ^P  + ^ e ) . Solving 3.3 for  p and substituting into 3.4, we obtain - ( 1 - p\)U\W  -P) + \(l-p+  U'(-)  = 0 (3.5) Equations (3.2) and (3.5) characterize the optimal contract.6 Our interest is in the impact on the equilibrium contract of  an increase in aggregate uncertainty. We represent an "increase in uncertainty" as a mean preserving spread in the conditional distribution of  losses, but with the further  restriction that A remains constant in this increase. That is, an increase in aggregate uncertainty is represented, as dH  > 0 with the restriction dL = —A/(l - A) • dH  . Totally differentiating  (3.2) and (3.5) with this substitution yields — (c + A(1 — c))dE + (1 — X)dP  + XpdH  — 0 ' (3.6) 6Note that if  c equals zero, so that we have a perfect  capital market, then (3.5) implies that the two marginal utilities are equal, which in turn implies full  insurance. This equation shows also that if  (3.5) is positive, then the coverage is less than full  in the bad state! A ( l - c ) V c + A(1 — c) (3.7) + (d  - p\) u" {w-P) + x ( i - P + IzLiru) c + A(1 — c) From (3.6) and (3.7) it can be shown that dE/dH dP/dH — A - l K1 + - A p where A is given by (3.8) ^ (I  - P + U"(-)  (1  - PX)U"(W  - P ) + X(L- P+ c+A ( l - c ) - ( A ( l - c ) + c) 1 - A Proposition 3.2: With  aggregate uncertainty in the size of  losses, an increase in uncertainty leads to (a) an increase in equity, E; (b) an increase in the premium, P; and (c) an increase in the equity-to-premium ratio, E/P. Proof:  We can write (3.8) in shorthand as with all of  the lower-case letters on the right-hand side positive. (This can be shown - - - l -dE/dH —a —b —e dP/dH - g  d - / using U"  < 0.) Solving for  dE/dH  gives dE/dH i — (ad+bg) -de — bf)  > 0, proving (a). Solving for  dP/dH  yields dP/dH  — _ ( n J + 6 g ) + «-/)• Substituting back in the terms for  (—ge + af)  yields (-ge  + af)  = (c  + X(l-c)) A ( 1 - P + r ~ y ] ~ C l ) U " ( . ) c+\(l-c) v V c + A(1 — c) J = (c + A(1 - c) - A2(l - c)) (l-p+ ^ w l ' f )  ^"(O c + A(1 - c) Therefore  (—ge  + af)  < 0, hence dP/dH  > 0, proving (b). To prove (c), re-write (3.2) as P c + A(1 — c) L E = 1 — A + P E from which we have p dH  E2 dL dE_ dH~  dH P_ E2 - E - ^ - L ^ -1 - A dH which is negative since dE/dH  > 0 by (b). Leverage is therefore  decreasing in uncertainty. QED 3.2.2 Uncertainty in Accident Probabilities Assumptions The alternative structure is one in which common factors  are in the events of accidents. We assume now that the loss from an accident is known, and equal to L, but, because of  dependence in the events of  accidents, the frequency  of  accidents is random. This frequency,  p, is assumed to take on two possible values, a and b, with b > a. The term A now represents the probability of  the frequency  b of  accidents. The ex ante probability of  an accident for  any individual is p = (1 — X)a  + Xb. A contract now involves the promise of  a payment I in the event of  an individual accident in exchange for  the premium P. In contrast to the case of  uncertain losses that are identical across individuals, where contractual promises for  cash flows  are always met, we must introduce here the notion of  bankruptcy. An insurer with equity-per-contract E is bankrupt if  P + E - pi < 0. We allow for  the possibility that bankruptcy involves the loss of  specific  assets, interpreted as a reputation for prudence, or other bankruptcy costs. As before,  issuing equity requires a transaction cost of  c per unit. As before,  we consider the contract offered  by a competitive market to identical, risk-averse consumers. This is the contract that maximizes individual expected utility subject to a zero-profit  constraint. Equilibrium Depending on the market parameters, especially the size of  bankruptcy costs and A, the equilibrium may or may not involve bankruptcy in the event that the acci-dent frequency  is b. In the case where bankruptcy costs are sufficiently  large, the equilibrium contract in this model will satisfy  the solvency constraint in both states. We consider this a reasonable approximation, in light of  the regulatory solvency con-straints faced  by firms.  These constraints do not, evidently, reduce the probability of  bankruptcy to zero; but the rate of  bankruptcy is very small with less than one percent of  policies defaulted  on in any year. In understanding the costs and benefits in the choice of  an equity ratio by a firm facing  existing solvency regulation, and generating testable implications regarding this choice, approximating the regulation as a complete constraint against bankruptcy is useful. 7 The expected net profit  to shareholders from the policy (P, I)  with equity E is —cE + P — [(1 — A)a + A6] I.  When the firm is subject to a no-bankruptcy condition for  both events, a and b, the gross return to shareholders in the event b is zero, since (it is easily shown) excess equity will not be issued. The amount of  equity, E, will be chosen given the contract (P, I)  to meet the no-bankruptcy constraint in event b: that the amount of  equity remaining after  payment of  the costs cE, covers net losses: (1  — c)E + P > bl. Let p = (1 — X)a  + Xb.  The equilibrium contract is characterized by the maxi-mum of  expected utility subject to the no-bankruptcy constraint and the zero profit constraint: max(l-p)U(W-P)+pU(W-P-L  + I)  (3.9) subject to (1 -c)E-+P>bI  (3.10) -cE + P-  [(1 - X)a  + Xb}I  = 0 (3.11) 7We have elsewhere considered in more detail the effect  of  actual solvency regulation on insurance markets (Winter (1991)). Proposition 3.3: In  the case of  uncertain probabilities with a no-bankruptcy constraint, -(a) an increase in uncertainty leads to an increase in E if  uncertainty is sufficiently small. That  is, dE/db\p > 0 if  b — a is sufficiently  small. (b) For  larger levels of  uncertainty, dE/db may be positive or negative. (c) For  utility satisfying  U'(W)  > 0, U"(W)  < 0, optimal equity is increasing in the degree of  risk aversion 7  = • Proof: Solving the first  (no-bankruptcy) constraint for  I yields I  = (l-c)E  + P (3.12) and substituting it into the zero profit  constraint yields c + |( 1 - c ) E+[1-^)P  = 0 (3.13) Let A = c+Ul-c) (3-14) Solving (3.13) for  P  and substituting this value into the objective function  in (3.9) yields the following  as a characterization of  the optimal equity: max(l - p)U(W  - AE) + pU  (w  - L + E (3.15) Letting W^A  and WA  be shorthand for  the realized wealth in the no-accident and accident states respectively, and setting the derivative of  this expression with respect to E to zero gives This is the first  order condition. Since \imb_>pA — 0, this derivative is unbounded as b —> p. Therefore,  for  b sufficiently  close to p (i.e. for  sufficiently  small uncertainty) the optimal E is positive. However, for  zero uncertainty (b  — p), the optimal contract is easily shown to yield the standard full  insurance solution: I  — L, P — pL and E — 0. It follows  that, holding p constant, dE/db > 0 for  b sufficiently  close to p. To prove (b), a parametric example suffices.  Define  the form of  the utility function as U(W)  — —e~lW,  and define  parameter values of  {p,  c, 7, L, W}  — {.1, .1,1, .25,1}. Optimal equity of  E*, defined  as being the value of  E which satisfies  (3.16), can be evaluated as b changes. The following  diagram plots optimal equity given changes in b as it ranges from .101 to .300: (1 - P)AU\W na) + p [1—? + ( i - l)>ll U'(W A) = 0 b b (3.16) 0.002 0.014 0.012 0.004 0.008 0.006 0.01 0.15 0 . 2 0.25 0 . 3 Figure 3.1 - Optimal equity under changing uncertainty Initially, E* increases as b increases, reaching a maximum where b = 0.2214. Beyond this point, E* is decreasing as uncertainty in the probability of  loss increases further. To prove (c), consider the first  order condition, (3.16). First, note that because U'(W)  > 0 and U"(W)  < 0, it is true that U'(x)  < U'(y)  where x > y. It can be shown that (1 - p ) A > p + (± - 1 )A]. For (3.16) to hold, it must be the case that U'(W N A) < IJ'(W A), meaning that W N A > W A. Wealth is greater in the no-accident state, which means that in equilibrium there is less than full  insurance. Given that there is less than full  insurance, consider how increasing risk aversion affects  the optimal level of  equity, E*. Define  the measure of  risk aversion as 7 = ~mw)- n o t e d i n ^ e previous paragraph, it is true that, x > y, U'(y)  — U'(x)  > 0. It is further  the case that > 0. One can define  (WNA,  WA)  as satisfying (3.16) for  some level of  risk aversion 7 , and (W N A,WA)  as satisfying  (3.16) for  a higher level of  risk aversion 7. It must be the case that WNA-W a<W N A-W A (3.17) since the higher degree of  risk aversion 7, combined with the requirement than (3.16) hold, implies that the difference  between the no-accident and accident wealth levels must be less than for  the lower risk aversion case. Put another way, as risk aversion increases, the optimal contract moves closer to full  insurance. Substituting into (3.17) for  {W N A,W A,W N A,W A}, and defining  {E*,E*} as the optimal levels of  equity for the two different  levels of  risk aversion, we have 1 - c b 1-c (3.18) (3.19) (3.19) can be simplified: The left  term of  (3.20) is positive, meaning that \JE* — E*j is negative. Therefore, optimal equity for  the higher degree of  risk aversion, E*, is greater, meaning that optimal equity is increasing in the degree of  risk aversion. QED At the heart of  the comparative statics in proposition 3 are two off-setting  effects of  an increase in uncertainty on equity. Holding constant the amount of  coverage issued, / , an increase uncertainty b implies an increase in the value of  equity, E, that is necessary to cover the claims at a given premium. This is the input effect. The amount of  coverage will drop, however, as a consequence of  the higher cost of  offering  any amount of  coverage; this feeds  back to a decrease in E: an output effect.  When uncertainty is sufficiently  low, the input effect  dominates and when uncertainty is high, the output effect  may dominate. The two effects  can be seen in the total differentiation  of  the no-bankruptcy condition, which yields dE/db — i+b(di/db)—dP/d b ^  r^^ t w Q t e r m s 0 f  this are, respectively, the input effect  and the output effect.  Endogenizing the change in P  through total differentiation  of  (3.11), holding p constant, yields d E r / , again showing a decomposition into the input and the output effects. Proposition 4: In the case of  uncertain probabilities with high bankruptcy costs, an increase in uncertainty leads to an decrease in the leverage ratio P/E.  ' Proof:  Solve (3.13) for  ^ = A, from which Hi)  _ P db  (b-p) 2 The ratio of  liabilities to equity is therefore  decreasing in uncertainty. QED To summarize the main comparative static results that flow  from the model: in-creasing aggregate uncertainty leads to an increase in optimal amount of  equity when the uncertainty is in the size of  the loss (conditional upon an accident) but a non-monotonic relationship in the case where the uncertainty (i.e. dependence) is in the events of  accidents. In both cases, the equity to premium ratio is increasing in un-certainty. Since the premium is the market valuation of  an insurer's liability (this liability being of  course the promise of  insurance payouts), the inverse of  this ratio is the liability to equity ratio, analogous to the debt-equity ratio conventionally used to summarize capital structure. The negative relationship between the liablity-equity ratio and the level of  uncertainty is the first  implication of  the theory that we will test in the next section of  this paper. Extension: Initial Equity Endowment An additional implication follows  from a simple extension of  the model.. Let E* represent the optimal equity that would be issued in a competitive insurance market under either set of  assumptions that we have set out in the model. Suppose now that firms  are endowed with an amount of  internal equity, A, at the beginning of the period on which they do not have to incur issuance costs. This endowment represents internal capital inherited from retained profits  earned previously. (We retain the static model assumption that the equity of  the firm is distributed entirely to shareholders at the end of  the period.) Then the equilibrium amount of  equity, E, in the extended model is E — max(E*,  A): if  A < E*, the equilibrium will be identical to the equilibrium analyzed in the model above. Those firms  endowed with substantial internal equity will earn rents on this endowment, but the contract will reflect  the opportunity cost of  capital at the margin, including the issuance costs. For these values of  A, dE/dA = 0. On the other hand, when A exceeds E*, then the entire equity is retained until the end of  the period and dE/dA = 1.8 It is clear that over the region A > E*, the premium P  is non-increasing in A; P  is decreasing in A over the subset of  this region where the "limited liability constraint is binding. It therefore  follows  that over this region d(E/P)/dA  > 0, or d(P/E)/dA  > 0. The prediction is that past profitability  should have a negative impact on leverage. 8Allowing firms  to distribute internal equity (through a special dividend or share repurchase) at the beginning of  the period does not change the essential results. 3.3 Evidence 3.3.1 Introduction Cross-sectional data on a sample of  U.S. property-liability insurers allows us in this section to provide evidence on two aspects of  insurers' capital structure decisions. The first  is the implication from our model that insurers' leverage should be decreasing in the uncertainty faced  in predicting average risks. While the equilibrium equity in our model is for  particular cases non-monotonic in uncertainty, leverage - as measured by the ratio of  insurance revenue to equity - is unambiguously decreasing in uncertainty. The second aspect of  capital structure on which we offer  evidence is the relative costs of  internal and external equity. The extension of  our model to include an endowment of  low-cost equity suggests that firms  with greater access to less costly internal capital will use less leverage. Recent theory on the economic dynamics of insurance markets relies on the assumption that internal capital is less costly than external equity. By a cost advantage to internal capital, we mean simply that there is a positive cost to the round-trip of  distributing an amount of  cash to equity holders, then raising the same amount through the issuance of  new equity. (The basis for such a cost is well-developed in the literature, e.g. Myers and Majluf  (1984)). Up to now, this assumption has been tested for  insurance markets using the time series of insurance market pricing. The implication of  this assumption for  the cross-section is i that leverage should be decreasing in recent profitability,  since this profitability  leads to greater accumulation of  internal equity.9 3.3.2 Empirical Proxies and Estimation The firm specific  data are collected from A.M. Best's Aggregates and Averages annual reports on consolidated property-casualty insurance companies. These statu-tory financial  information  are filed  by insurance companies to National Association of  Insurance Commissioners (NAIC) to assist insurance commissioners in regulating and monitoring insurance companies licensed in their respective state. The selected sample covers 852 U.S. property-casualty stock insurance companies from 1999 to 2004. Mean Std. Dev. Minimum Maximum NPE/E 1.0455 0.6814 0.0003 4.5491 SD 0.1946 0.4310 0.0097 4.8975 SIZE 1,147,510 3,837,997 769 50,959,623 PROFIT 0.0491 0.2080 -4.0868 1.5836 Table 3.1: Descriptive Statistics The cross-sectional regressions of  firm capital structure (leverage) on three hypoth-esized determinants— uncertainty of  insurance loss, firm size, and past profitability, are specified  as 9The tests of  both hypotheses for  insurance markets are parallel to tests of  capital structure hypotheses for  general corporations that have been offered  in the financial  economics literature (e.g., Bradley, Jarrell and Kim (1984), Titman and Wessels (1988)). log (NPE/E)  = <* + /?• log (SD)  + 5 • log (SIZE)  + 77 • PROFIT  + e (3.21) where NPE  is Net Premiums Earned, E is the Policyholders Surplus, SD is the uncer-tainty of  the insurance loss, SIZE  is the firm size, and PROFIT  is past profitability. These empirical proxies are defined  as follows. • Policyholders' Surplus E\ the equity of  a property-casualty insurance firm. • Net Premium Earned NPE:  the total insurance policy revenue from policies issued during a given year, adjusted for  any increase or decrease in liabilities for  unearned premiums during the year.10 • Loss Ratio: the ratio of  incurred losses and loss adjustment expenses to net premium earned. • Capital Structure (NPE/E):  is measured as the ratio of  Net Premium Earned (NPE)  to Policyholders' Surplus (E)  in 2004 This reflects  the relationship between the current volume of  net insurance liability and the equity. • Uncertainty of  the insurance loss SD: is represented by the standard deviation of  the loss ratio from 1999 to 2004. The theoretical model predicts an inverse relationship between the capital structure and the uncertainty in insurance market. • Firm Size SIZE:  the costly external equity suggests that it is more difficult  for 10Net Premiums Earned record premium income for  the year, prorated for  the portion of  the policy that occurs during the year in question. This is not a perfect  proxy for  liabilities, particularly in the case where policies are written on an occurrence basis. Where the firm has written occurrence policies in the past, the premiums have.already been earned, but the liability still exists in that claims may still occur in. the future. smaller firms  to issue equity in times of  increasing of  aggregate uncertainty; therefore, smaller insurance firms  should tend to keep a higher equity-liability ratio. Warner (1977), Ang, Chua, and McConnel (1982) and Titman and Wessels (1988) provide evidence for  non-financial  firms  that capital structure is related to firm size. One explanation for  this is that transaction costs are decreasing in the size of  the firm. Smith (1977) finds  that small firms  incur substantially more costs to issue equity than large firms. 1 1 In the regression, the natural logarithm of  total admitted assets is used as a proxy for  firm size. The predicted sign in the regression is positive. • Past Profitability  PROFIT:  A positive cost of  issuing equity, or a positive cost of distributing cash to shareholders implies a negative relationship between the capital structure and past profitability.  This is because this positive cost of  equity implies that the internally generated funds  are low-cost source of  equity capital for  the insur-ance firm.  The sample average of  the profit/surplus  ratios from 1999 to 2004 is used as a proxy of  firm's  past profitability. Although there are 852 firms  in our sample, a number of  these firms  are part of the same insurance group. Our sample contains 345 unique insurance groups. It is reasonable to assume that there will be some correlation of  errors within each group. We correct for  this by using the Huber-White sandwich estimator, which provides 1 1 The transaction costs of  issuing securities are defined  as flotation  costs and costs encountered in trying to secure the highest price for  the firm's  securities. Smith (1977) identified  flotation  costs as: (1) compensation paid to investment bankers, (2) legal fees,  (3) accounting fees,  (4) engineering fees,  (5) trustee's, fee,  (6) listing fees,  (7) printing and engraving fees,  (9) federal  revenue stamps, and (10) state taxes. Smith went on to provide evidence which showed that firms  enjoy economies of  scale when issuing securities. robust standard errors 3.3.3 Results The results are reported in Table 3.2. DEPENDENT VARIABLE: NPE/E BP = 0.3218 Coefficient Robust Std. Error t P>|t| SD -0.708* 0.067 -10,55 0.000 SIZE 0.680* 0.023 2.98 0.003 PROFIT -0.638* 0.205 -3.12 0.002 Intercept -3.214* 0.489 -6.57 0.000 Table 3.2: Results The estimated elasticity of  leverage with respect to uncertainty (SD)  is -.708, which is both statistically and economically significant.  This result confirms  our hypothesis that leverage is indeed decreasing in the variance of  firms'  loss ratio. Firms faced  with more uncertainty do choose capital structures which use less leverage. The coefficient  on past profit  (PROFIT)  is -.638, which is also statistically and economically significant.  This confirms  our second hypothesis, which is that firms that have greater access to internal capital (in this case, due to recent profitability) tend to use less leverage than do firms  with a lesser supply of  internal capital. This supports the notion that there is a cost advantage to internal equity, which is at the heart of  previous studies of  the behavior of  insurance markets. Finally, the coefficient  on our variable controlling for  SIZE  is 0.680. Larger firms use greater leverage than smaller firms,  as predicted. 3.4 Conclusion This paper explores the capital structure of  insurers. The focus  is on the impact of  aggregate uncertainty, or dependence among risks, since this is the source of  an insurer's incentive to issue equity. Insurance firms  respond to the shocks of  increased risks by taking all or some of  the following  actions: placing limits on the number or coverage of  contracts that they offer;  raising premium for  the policies that they issue; and raising more equity. We analyze the equilibrium mixture of  these responses in a competitive insurance market, and find  that the impact of  increasing uncertainty on the equity decision depends on the nature of  aggregate uncertainty. Where this uncertainty is in the size of  losses, equity increases with uncertainty; where the risk dependence is in the events of  losses, equity first  increases then decreases with un-certainty, providing that individuals are not too risk averse. The latter result follows from a tradeoff  between two effects,  which we label the input effect  of  uncertainty, and the output effect.  In both cases, however, the ratio of  equity to insurance rev-enue increases. We extend the model to look at the effect  of  a cost difference  between internal equity (less costly) and external equity (more costly). This extension leads to the hypothesis that firms  with greater internal equity will tend to use less leverage. We test both hypotheses directly on a sample of  852 U.S. property and casualty stock insurers over a sample period from 1999-2004. We find  support for  both of our hypotheses. Firms that have higher variance in their loss ratio, our proxy for uncertainty, use significantly  less leverage, supporting our theory that uncertainty and leverage are negatively correlated. Firms that have been recently profitable, implying greater internal capital, use significantly  less leverage. This supports the theory that there is a cost advantage to internal over external equity, which is at the core of  recent theories of  insurance market dynamics. 3.5 Bibliography [1] Ang, J., Chu J., and J. McConnell, 1982, "The Administrative Costs of  Corporate Bankruptcy: A Note", Journal  of  Finance  37, 219-226. [2] Arrow, K.J., 1971, "The Role of  Securities in the Allocation of.Risk-Bearing", Essays in the Theory  of  Risk-Bearing, Chicago, Markham. [3] Best's Aggregates and Averages, annual 1980-1992. [4] Bradley, M., Jarrell, G.A., and E. Han Kim, 1984, "On the Existence of  an Opti-mal Capital Structure: Theory and Evidence", Journal  of  Finance  39, 857-78. [5] Cummins, J.D., and S. Harrington, 1987, "The Relationship between Risk and Return: Evidence for.  Property-Liability Insurance", Journal  of  Risk and Insurance, 54. [6] Doherty, Niel A., and Harris Schlesinger, 1990, "Rational Insurance Purchasing: Consideration of  Contract Nonperformance",  The  Quarterly Journal  of  Economics, 243-253. [7] Fischer, E., Heinkel, R. and J. Zechner (1989), "Dynamic Capital Structure Choice: Theory and Tests", Journal  of  Finance  44, 19-40. [8] Gr0n, A., 1994, "Capacity constraints and cycles in property-casualty insurance markets", Rand Journal  of  Economics 25, 110-127. [9] Jensen, Michael C., and William H. Meckling, 1976, "Theory of  the Firm: Agency costs, Managerial Behavior and Ownership Structure", Journal  of  Financial  Eco-nomics 4, 305-360. [10] Marshall, John M., 1976, "Insurance Theory: Reserves Versus Mutuality", Eco-nomic Inquiry,  476-492. [11] Myers, Stewart, 1984, "The Capital Structure Puzzle", Journal  of  Finance  39, 575-92. [12] Myers, Stewart, and N. Majluf,  1984, "Corporate Financing and Investment Decisions When firms  have information  investors do not have", Journal  of  Financial Economics 13, 187-221. [13] OneSource, "U.S. Insurance and Reinsurance Transactions", 1-800-554-5501. [14] Rajan, R.G. and L. Zingales, 1994, "What do we know about capital structure? Some evidence from international data" (Graduate School of  Business, University of  Chicago). [15] Shyam-Sunder, Lakshmi, and Stewart C. Myers, 1994, "Testing Static Trade-off Against Pecking Order Models of  Capital Structure", NBER working paper, No. 4722. [16] Smith, C., Jr, 1977, "Alternative Methods for  Raising Capital: Rights Versus Underwritten Offerings"  Journal  of  Financial  Economics 5, 273-307. [17] Titman, S., 1984, "The Effects  of  Capital Structure on a Firm's Liquidation Decision", Journal  of  Financial  Economics 13 (March 1984), 137-51. [18] Titman, Sheridan and R. Wessels, 1988, "The Determinants of  Capital Structure Choice", The  Journal  of  Finance  43, 1-19. [19] Warner, J., 1977, "Bankruptcy Costs: Some Evidence", Journal  of  Finance  32, 337-347. [20] Winter, Ralph A. 1988, "The Liability Crisis and the Dynamics of  Competitive Insurance Markets", Yale  Journal  on Regulation 5, 455-550. [21] Winter, Ralph A. 1991, "Solvency Regulation and the Insurance Cycle", Eco-nomic Inquiry. [22] Winter, Ralph A. 1994, "Dynamics of  Competitive Insurance Markets", Journal of  Financial  Intermediation. Chapter 4 Contracting With Agents of Heterogeneous Risk Aversion 4.1 Introduction \ • • -i 4.1.1 Overview Firms' shareholders hire managers to look after  their interests. Managers' tasks can be crudely divided into two categories. The first  task is project selection, where managers decide what lines of  business the firm ought to pursue, and what invest-ments ought to be made. The second task is effort  exertion, where managers can improve the distribution of  eventual project outcomes by working harder. It is im-possible for  shareholders to know whether managers' actions were optimal from their perspective on either of  these tasks. With respect to project selection, the manager has an informational  advantage that comes about, either because the manager was specifically  hired for  his expertise in this area, or because his position affords  him the opportunity develop a better knowledge of  the firm's  opportunities than anyone else. It is impossible for  shareholders to know whether the manager's project selec-tion decision was the "right" one. With respect to effort  exertion, it is assumed that shareholders simply cannot monitor the manager's level of  effort. Dealing with these problems is the standard purview of  the principal-agent liter-ature. The solution, particularly with respect to motivating the proper effort  level on the part of  the manager, is to make part of  the manager's pay package depend on firm performance.  The advantage of  incentive pay is that it aligns shareholder and managerial interests; the disadvantage is that it involves a deviation from optimal risk sharing. Well diversified  shareholders are presumed to be risk neutral with respect to the firm's  idiosyncratic risk. Risk averse managers are unable to diversify  their expo-sure to firm risk, meaning that they place less value on risky pay than it is expected to cost the firm's  shareholders to provide it. The standard approach to determining the optimal contract is trade off  the costs and benefits  of  incentive pay, choosing the level of  power (i.e. amount of  incentive pay provided) at which the marginal costs equal the marginal benefits. I motivate this paper with a number of  observations. The first  is that managers differ  in their degree of  risk aversion, and that a manager's risk aversion is not observ-able. This complicates the problem of  choosing the correct tradeoff  between inducing managerial effort  and deviating from optimal risk sharing. Second, higher managerial risk aversion is costly in two ways. First, higher risk aversion means that the manager puts a lower value on risky pay. This implies that the cost of  motivating effort  exertion is increasing in managerial risk aversion. How-ever, managerial risk aversion is also costly in terms of  motivating correct project selection. When selecting projects, managers have an incentive to choose those that best fit  their own interests, as opposed to those of  firm shareholders. This becomes important when projects differ  in dimensions such as the degree of  risk they impose. The greater difference  in risk preferences  between risk averse managers and risk neu-tral shareholders, the greater will be the distortion imposed by managers selecting projects according to their own interests. As such, shareholders prefer  managers with lower risk aversion for  two reasons: it is less costly to motivate effort  exertion, and these managers' project selection decisions will more closely match shareholders' preferred  outcomes. A third observation is that the market for  managerial labour, like any labour market, is a competitive one. Firms compete with one another for  the services of preferred  managers, and managers will choose to work for  the firm that makes them the offer  they prefer. These three observations taken together imply the following.  From the second observation, it is clear that firms  prefer  lower risk aversion managers. From the third observation, they must compete against other firms  for  the services of  lower risk aver-sion managers. And from the first  observation, such competition is difficult,  since a manager's risk aversion is his own private information.  Firms must therefore  develop contracts which serve as screening devices, designed so that they will attract low risk aversion managers. Since all managers prefer  more pay to less, firms  cannot compete for  low risk aversion managers simply by raising wages. If  they wish to separate relatively desirable low risk aversion managers from relatively undesirable high risk aversion managers, they must compete in a manner that exploits the differences  be-tween types. Since lower risk aversion managers put greater value on risky pay than high risk aversion managers, firms  have an incentive to offer  high-powered contracts as a screening device. Such contracts appeal to the targeted low risk aversion managers, but not to high risk aversion managers.1 This paper explores the impact that this selection effect  has on the design of  man-agerial contracts. I develop a model where firms  must compete against one another in the managerial labour market to attract managers who are responsible for  both project selection and effort  exertion. In this setting, incentive contracts perform two functions.  The first  is to serve the traditional role of  motivating the correct effort choice. The second is to act as a screening mechanism, helping firms  compete for 1 Screening models are more traditionally thought of  in the context of  insurance contracts, where an insurer sets out a menu of  contracts to offer  to customers who walk in the door. The practice of  hiring a CEO is clearly a much more selective one, and firms  put enormous effort  into learning as much as possible about prospective candidates. However, for  any executive position the firm  will identify  a number of  candidates, whose risk aversion will, likely remain difficult  to discern ex ante. Similarly, candidates for  top positions generally appeal to more than one prospective employer. As such, a competitive screening model is a reasonable approach to modelling the problem. the services of  a lower risk aversion manager whose preferences  lead to better project selection. 4.1.2 Literature review Principal-agent problems have long been studied in the literature. Papers by Jensen and Meckling, Ross, Holmstrom, and Holmstrom and Milgrom are well-known early examples which highlight the difficulties  of  contracting between a principal and his agent when the agent's actions are not easily observed. The traditional principal-agent model developed in most of  these (and later) papers is one where an agent, by exerting costly effort,  is able to improve the expected outcome of  a principal. This is often  described as the agent exerting "productive effort".  A less common, but very interesting, related form of  model is based instead on the agent exerting effort  in order to evaluate a number of  potential projects. The principal must motivate the agent to expend effort  to examine several opportunities, and then implement one of them, with the principal unable to observe the agent's selection process or decision criteria. Lambert's (1986) model of  "evaluation effort"  is one of  the earlier models that capture this idea, which has also been modeled recently by Core and Qian (2002). Papers examining the compensation of  investment fund  managers often  use a similar structure. The early research into principal-agent problems has been taken up with enthusi-asm recently as efforts  to explain the nature of  executive compensation have grown almost as quickly as the compensation itself.  Murphy (1999), Hall and Murphy (2001) and Core, Guay and Larcker (2002) provide excellent surveys of  the voluminous lit-erature that has exploded around the general question of  why executives are being compensated as they are, and whether the compensation they receive is consistent with optimal contracting. An important research question is to ask how managers behave when provided with risky compensation. Numerous papers have argued that this type of  pay encourages risk taking. Taken to the extreme, consider risk neutral managers who are provided with a call option on an asset whose underlying volatility they control. Since the risk neutral value of  a call option is increasing in underlying volatility, the managers would choose as risky an investment strategy as they possibly could. This solution isn't very satisfying,  particularly because most of  the many managers compensated with share options do not seem to be trying to drive their firms'  volatility to unprecedented and dizzying levels.2 Several recent papers have introduced the notion that an executive's risk aversion should be considered when analyzing how risky pay will affect  his project or investment selection. Since the executive has a great deal of  wealth tied up in firm-specific  securities, in addition to his human capital being highly correlated with firm performance,  it's not unreasonable to believe that a manager will be very concerned about his firm's  idiosyncratic risk. Carpenter 2The Skillings and Enrons of  this world remain more the exception than the rule, although some might wish to debate this. (2000) finds  that a mutual fund  manager who is risk averse with respect to investment performance  will, in some circumstances, actually behave more conservatively if  given more options. Recently, both Ross (2004) and Lewellen (2003) explore how risk aversion filters  the effect  of  risky compensation on managerial decision making. These and other papers make the case that to understand the effect  of  any compensation package, a manager's private preferences  toward risk are a crucial element that must be considered. Within the context of  the executive compensation literature, relatively little has been done with regard to considering how differences  in agents' risk aversion affects the design of  pay packages. An exception is Jullien et al (2000), which considers executive compensation as one application of  their model describing how a risk neutral principal ought to contract with a number of  risk averse agents having heterogeneous, private levels of  risk aversion. Unlike this paper, their model assumes that the firm takes on numerous principal-agent relationships, as opposed to contracting with only one agent. Serfes  (2005) considers a matching game between risk neutral principals and agents of  differing  risk aversion in a labour market setting. However, in this model agents exert only productive effort,  and do not make project selection decisions. Wright (2004) presents a model similar to that in this paper, in terms of  a setting featuring  two types of  agents with firms  competing for  their services. Again, this model takes firm risk as exogenous, and agents make no project selection decisions. Both the Serfes  and Wright papers predict that agents of  lower risk aversion will contract with firms  that are a priori riskier. Understanding risk aversion, and differences  in risk aversion between agents, is im-portant when considering results from empirical compensation studies. The Jensen and Murphy (1990) result asserting that executive pay was "lower" than it ought to have been is well known. But interpreting the degree to which pay changes relative to changes in firm wealth must be taken in the context of  executive risk aversion. Haubrich (1994) attempts to measure this for  a single type of  agent, while this pa-per emphasizes that differences  in risk aversion and the associated constraints* on contracting should be considered as well. As well, there is some question as to why the relationship between the degree to which pay is risky and the level of  firm-specific  risk is so tenuous. A standard argu-ment in the literature is that if  executives are risk averse, then they should be asked to bear less risky compensation as firm risk increases. Prendergast (2002) surveys the empirical literature on this question and finds  the evidence is decidedly mixed (three studies find  the predicted negative relationship, three a positive relationship, and six no statistically significant  relationship). His paper argues that the ambiguity of  the contract setting is a contributing factor.  I suggest that differences  in risk aversion may also play a role. For example, risky high technology firms,  faced  with hiring a manager to select projects in an unstructured environment, have a strong incentive to bid for  low risk aversion agents. When contracting with low risk aversion agents, these firms  can take advantage of  the agent's risk tolerance by offering  high powered incentive contracts, despite the relatively high firm risk. Less risky firms  contracting with higher risk aversion agents would then not necessarily offer  higher powered con-tracts, despite these firms'  lower return volatility. Thus the relationship between firm risk and power of  incentive contracts would not necessarily be negative, as standard theory predicts. 4.2 Model: Single Firm, Single Agent Consider a firm with the opportunity to hire a manager to select a one-period investment project, and then exert effort  to implement the project. The manager will choose between a safe  project and a risky project, the safe  project returning a value of  vo, while the risky project's terminal value will be either vg or Vb-  The prior probability of  obtaining the high or low outcomes is 1/2 for  each. So that one project does not dominate the other ex ante, vb < vQ < vg. For ease of  exposition, the risk free  interest rate is assumed to be zero. The firm's  shareholders are well diversified,  and therefore  risk neutral with re-spect to the idiosyncratic risk posed by the uncertain project. Prospective managers' prospects for  diversification  are much less, and they are therefore  assumed to be risk averse with respect to an employment contract that calls for  their wage to have a stochastic component. Once hired, the chosen manager receives a signal r, which is the probability that the risky project will return vg. f  is distributed uniformly  over [0,1]. The signal is the manager's private information,  and he is unable to communicate this signal credibly to firm shareholders. With the updated signal, the manager chooses between the safe  project and the risky project. If  he chooses the safe  project, the firm's  terminal asset value is VQ  with certainty. The manager does not need to exert any effort  in this case. However, if the manager chooses the risky project, he has the opportunity to exert effort,  at a personal cost of  c dollars, to improve the probability of  success. If  the manager exerts effort,  the probability of  realizing the high asset value vg is r (the realization of  the manager's signal), while the low value vb will occur with probability 1 — r. If  the manager does not exert effort,  the project will certainly fail,  and the terminal value of  the project is vb with probability 1. The manager's effort  is not observable to firm shareholders. To provide the man-ager with an incentive to exert effort,  the firm must offer  the manager an incentive contract; that is, the manager's payoff  must depend on the firm's  terminal asset value. A contract S therefore  takes the form S = [so, sb, sg], defining  the payment that the manager receives for  each possible outcome in firm asset value.3 3When productive effort  is removed from the model and effort  is shifted  to the first  stage of  the game, the model collapses to that of  Lambert (1986). Lambert's solution differs  markedly from the joint selection-production model developed here. 4.2.1 First best solution If  the shareholders could observe the manager's signal, and determine whether the manager working on the good project exerted effort,  the first  best result would be possible. At the project selection stage of  the game, the investment policy taken by the manager can be described in terms of  a cutoff  point p. When the signal is below this point, the prospect of  the risky project succeeding is too low, and the shareholders would prefer  that the manager pursue the safe  project. When the signal is above p, the probability of  success is sufficiently  high, and the shareholders would prefer  that the manager pursue the risky project. In the full  information  case, the firm pays the manager a fixed  wage, ui, equal to the manager's reservation wage. Should the signal indicate that the risky project is worth pursuing, the firm pays the manager a bonus of  c to compensate the manager for  the effort  required to make the good realization possible.4 The first  best rule is to select the risky project if  the realization of  r is such that rvg + (1  - r)vb - c > vQ • Talcing the first  best cutoff  point p*F B as the value of  r which makes this an equality and rearranging yields Vo-V b + C • ' PFB  = — — ' 4.1 v„ - vb • 4Since both the signal and effort  are observable in this case, there is no principal-agent con-flict.  The manager pursues exactly the investment policy that the shareholders desire, and effort  is verifiable  and therefore  contractible. The strategy of  making the cutoff  point, and taking the risky project if  and only if  the realization of  r satisfies  r > p*F B, maximizes expected asset value net of  the expected cost of  effort. 4.2.2 Hidden information:  the decisions of  risk averse man-agers When the manager's signal and effort  level are not observable, the payment to the manager can depend only on the realization of  firm asset values. The contract must not only provide the manager with an incentive to exert effort  in the appropriate circumstances (i.e. when he selects the risky project), it must also elicit the correct project selection decision at the first  stage of  the game. While this would be simple if  the manager were risk neutral, in reality managers are risk averse. The problem becomes more complex when this risk aversion is taken into account. The manager is assumed to have negative exponential (CARA) utility of  the form U(w)  = —e~lw (4.2) where 7 is the coefficient  of  absolute risk aversion. This functional  form has the property that the manager's decisions will not change as his level of  wealth changes, which provides convenient tractability when working to solve both the firm's  and the manager's respective maximization problems (see Holmstrom and Milgrom 1987). The manager is assumed to have an outside option which provides a certain pay-ment of  w. The manager will therefore  not accept any contract providing ex ante expected utility of  less than U(w)  = U.  However, once the manager accepts the contract at the start of  the game, the outside option disappears.5 When offered  a contract S = [s0, sb, sg], the manager will choose the investment cutoff  p to maximize his own utility. That point is where the manager is indifferent between pursuing the safe  project and exerting effort  on the risky project: U(s 0) = (l-p)U(s b-c)+pU(s g-c) (4.3) This implies that on a given contract, the manager's chosen cutoff  point p is given by p = P(S)  = U { S o ) ~ U { S b ' C ) (4.4) 4.2.3 The optimal contract The following  expressions are useful  in the derivation of  the optimal contract. Re-call that the updated signal of  risky project's probability of  success, f,  is distributed uniformly  over [0,1]. However, the ex ante probability of  the ultimate outcome be-ing the successful  risky project depends not only on the realization of  the updated signal, but also on whether or not the signal is greater than the cutoff  point p. Since contracting decisions are made prior to the transmission of  the updated signal, it is 5This rules out a strategy where the manager waits to observe the signal and quits if  it is not to his liking. helpful  to derive ex ante probabilities for  each of  the three outcomes (safe  project VQ. successful  risky project vg, and unsuccessful  risky project vb) given a cutoff  point p. Since the signal r is distributed uniformly  over [0,1], and the safe  project is pursued for  any realization below p, the probability of  pursuing the safe  project is F(p)  = p, where F  is the uniform CDF of  f.  The other two outcomes first  require that the signal be greater than p. The ex ante probability of  realizing the good distribution i conditional on the project's signal meeting the cutoff  point is J rf(r)dr  — - ( 1 — p2). p A similar argument can be made to show that the prior probability of  realizing the bad return is ^(1 — p)2. Let V(p)  be the expected value of  the firm's  assets under investment policy p, and C(S;p) be the expected wage cost of  offering  contract S when the manager's investment cutoff  policy is p. The firm's  objective is to maximize expected profit, 7r(S-,p), where 7r(S;p) = V ( p ) - C ( S ; p ) (4.5) and V(p)  = pv0 + \(l-p 2)vg + ^(l-p) 2vb (4.6) C(S;p) = pSo + ±(l-p 2)sg + ^(l-p)s b (4.7) where the probabilities associated with each outcome are the ex ante probabilities of the outcome occurring given the investment policy p. The maximization problem is subject to the following  constraints: pU(s 0) + ^(l-p 2)U(s g-c) + ^(l-p) 2U(s b-c) > U  (4.8) (1 -p)U(s b-c)+pU(s g-c) > U(s b) (4.9) (1 — p)U(s b — c) +pU(s g — c) = U{s 0) (4.10) Equation 4.8 is the manager's incentive compatibility constraint, requiring that ex ante the manager's expected utility from pursuing the investment policy p under the contract meet his level of  reservation utility. Equations 4.9 and 4.10 are the' individual rationality constraints which govern behaviour at the second stage of  the game, after  the manager receives the signal. Equation 4.9 requires that the manager prefer  to work under the contract than simply select the risky project and exert no effort.  Equation (4.10) requires that the manager not prefer  to exert effort  under the risky project rather than avoid effort  by selecting the safe  project when the signal is equal to the cutoff  point.6 4.2.4 Solution properties Proposition 4.1 In  the single agent, single firm  case, 4-8 binds in equilibrium. Proof:  Equation 4.8 is the incentive compatibility constraint, and holds that the manager's expected utility upon taking the contract must meet his level of  reservation 6There are a continuum of  incentive compatiblity constraints with respect to the payout from the safe  project, one for  each realization of  the signal f.  In equilibrium, all are redundant except for  4.10, where r = p. 4.10 is a rearrangment of  the condition from 4.4 requiring that the manager's chosen cutoff  point p satisfy  P(S). utility. Under negative exponential utility, a utility function  of  the Constant Abso-lute Risk Aversion (CARA) class, the decisions the manager makes with respect to investment and effort  decisions depend on the relative differences  between So, Sb and sg, not on their absolute levels. Consider any contract S,motivating cutoff  point p — P(S),  for  which the incentive _ / * compatibility constraint 4.8 does not bind. There must always exist a contract S = S — e, under which e is subtracted from the payment in every state made under the original contract, e is sufficiently  small such that 4.8 remains satisfied.  Taking 4.4 and substituting U(w)  = —e~lw, both contracts motivate the same decision p — P(S)  = P(S): _e-7(so-e) _ (_e-7(sf,-c-e)) P(S) ((—e-"Ks<>-c)) — (—e~T(Sb~c))) e - f ( - e ) = P(  S) Then it must be that tt(S-p)  < 7r(S;p) as the expected asset value is the same under each contract, while the expected wage cost is lower under S. S cannot be optimal. QED Proposition 4.2 At an optimal contract S, p = P(S)  > p*F B. Proof:  Define ;p) = pU(s 0) + ±(l-p 2)U(s g-c) + ±(l-p) 2U(s b-c) for  S = (s 0,sg,sb) A A * Consider a contract S such that P(S) = p < p*F B. Consider a deviation from S given • by S = (so, sg, sb) — (s 0 + 2 ( 1 ^ sg — e, sb). Then evaluate ^ for  S under the old p. This expression is greater than it was under S: vD(S;p)<*(S;p) since S is a mean preserving spread of  S.7 Because P(S)  ^ P(S),  the agent's equi-librium investment policy under S changes and is given by p — P(S). Because p is utility maximizing under S Define  t such that = V(s 0-t,sg-t,sb-t-p) Then 7r(S;p) = 7 r ( S ; p ) < 7 r ( S - t ; p ) The first  and second terms are equal since both are evaluated at p and the spread 7Note that while \I/(S;p) can be evaluated for  any p, it is not necessarily the case that p satisfies P(S).  However, for  ^(Sjp) to represent the true equilibrium expected utility from S, p must satisfy p = P(S). from S to S is mean preserving.8 P{S)  > P(S) because the individual rationality constraint (4.9) is satisfied  for  fewer  realizations of f  under S than under S. P(S)  < p*F B and P(S) < P(S),  therefore  V(p)  < V{p).  For e sufficiently  small, all surplus from the move from  (S,p)  to (S — t,p) accrues to the ^ • A A firm.  Thus given any contract P(S)  < p*F B a Pareto dominant contract exists, and S cannot be optimal. QED ' Proposition 4.3 In  the single agent, single firm  case, 4-9 binds in equilibrium. Proof:  Consider a contract S = (s0, sg , Sf>)  for  which 4.9 does not bind. Let > ^ p = P(S).  For small e, define  S such that \ 2 So = So - ; e V sg ~~ s9 h '= sb + e '  ^ . so S is a mean preserving spread of  S given p. Then . tt(S\p)  - 7r(S-p)  _ but 8Note that 7r(S,p) cannot be an equilibrium level of  profit  because the expression is evaluated at p and p ^ P(S). . because the risk averse agent prefers  to avoid the lottery presented by the mean A preserving spread in S. Because p is not a utility maximizing investment level for contract S, since P(S) = p, we have Define  t such that and note that p = P(S)  = P(S  - t) under CARA utility. Finally, 7 r ( S - t ; p ) > 7 r ( S ; p ) = 7r(S ;p) because p*F B < p < p, and the downward shift  of  t in C allows the principal to capture the surplus. Therefore,  when 4.9 does not bind for  S, there exists a Pareto dominant contract that leaves the agent's utility unchanged and makes the principal strictly better off.  QED Corollary 4.1 At the optimum s 0 = sb. That is, the wage paid for  selecting the safe  project is set equal to the wage paid when the risky project is selected and the bad return is realized. Proof:  Equation 4.9 binds and 4.10 is an equality. Both are identical on the left hand side, so U(s b) — U(s 0). By monotonicity of  the utility function,  sb — s0. QED Since equations 4.8 through 4.10 bind, it is possible to determine the minimum cost contract to motivate a given investment policy p. Substitute Sb — s 0 into 4.4: Define  g — sg — s0 , and since U(w)  = —e~yw, one can factor  out e _ 7 S ° and write „ Pfw U(0)-U(-c) Solving this expression for  g yields cy — Ln 1—ec7(l—p) g = G(p)  = 1 ? i (4.13) • • 7 . Define  w = s0 as the fixed  wage for  a given contract S. For a given g. the firm chooses w so as to strictly satisfy  the incentive compatibility constraint(4.8). The dimension of  the contract space is reduced to a fixed  wage w, and a bonus g paid when the good state is realized. I abuse notation by retaining S to denote (now) two dimensional contracts. Efficient  contracts therefore  take the form S = (w,  g), where g is the bonus that motivates the agent to pursue investment cutoff  policy p, and w is the wage required to meet the agent's reservation utility strictly. The expected wage cost of  motivating an investment cutoff  point p is C(S ]P) = w+(l-p 2)g where g — G(p).  It is therefore  possible to calculate the expected wage cost, C(S;p), of  motivating a given agent to pursue any cutoff  point p. as well as the expected asset value, V(p),  from the cutoff  point, p. The firm then chooses the contract S which motivates the cutoff  point p, at which point the marginal expected wage cost of  a change in p is equal to the marginal increase in expected firm value, ^ = QED Lemma 4.1 The  maximum expected utility (measured  in terms of  a certainty equivalent payment) an agent can derive from  any contract is decreasing in the agent's risk aversion. Proof:  Let be the expected utility that an agent of  risk aversion 7,, derives from contract S evaluated at p. ^(S;  p) = pUi{w)  + P*)Ui{w  + g~c) + ^(  1 - p)2U t(w  - c) Let CEii^i) be the certainty equivalent of  any expected utility for  an agent of risk aversion 7 Consider an agent L having risk aversion 7 L , and an agent H  having higher risk aversion = 7 L + e. Consider the expected utility either type derives from contract S by pursuing investment policy pn = P//(S), where pn is the high type's utility maximizing investment cutoff  point under the contract. Using the fact  that the agents' utility functions  are of  the form Ui = — e~7i : = p e - ^ w ) + h i - p 2 ) e - ^ w + 9 ~ c ^ + i ( l - p ) 2 e - ^ w ~ c ) i This is equivalent to the utility from a fixed  payment w and a lottery paying — c with probability |(1 —p)2, 0 with probability p and (g  — c) with probability \{l—p 2). Each agent places the same value on the certain payment w, and the difference  in certainty equivalent utility derived under the contract depends on the value each places on the lottery. Because H  is more risk averse than L, UH(X)  is a concave transformation  of UL(X).  Then it must be the case that L is willing to pay more than H  for  any lottery. Therefore CEL(Y L(S;p H))>CE H(Y H(S-,p H)) To complete the proof,  observe that while type H  pursues his expected utility max-imizing investment policy pH  = PH(S),  the policy is not utility maximizing for  L, whose utililty maximizing policy pL satisfies  pL — PL(S)  Therefore It follows  that CE l{V l{S-p L) > CEL(V L(S;p H))  > CEH^H{S;p H)) which proves the lemma.9 QED Proposition 4.4 The  expected wage .cost of  motivating any investment policy p < 1, is increasing in 7, the manager's risk aversion. Proof:  Consider an agent L having risk aversion and an agent H  having higher risk aversion 7# = 7^ + e. Both demand the same certainty-equivalent wage, 9If  p were fixed,  this lemma would be simply a representation of  one of  the outcomes of  increasing risk aversion described in Rothchild Stiglitz (1970). It is the endogeneity of  p that makes this result non-trivial. W.  Let SL = {WL,9L)  be the least cost contract that motivates type L to pursue investment policy p, and let SJJ  — (w H,  9N)  be a contract motivates type H  to pursue p. To prove the proposition,'consider a contradiction: C(S#;p) = C(S L-,p). From (4.14) (4.15) Define  the difference  in cost under the two contracts as AC = C(S H;p)-C(S L]p) = 0 (4.16) Since AC = 0 if  there is no difference  in cost, then Aw=l-{l-p 2)Ag (4.17) SH  is therefore  a mean preserving spread of  S^. Because the agents are risk averse, •VH(SL-,P)  >*„{S H ]p) (4.18) Consider the utility of  agent H  under SL. Let p = P f-[(S L) be the utility maximizing investment cutoff  for  agent H  under SL. Because p is not utility maximizing for  type H  under the contract ^ ( S z , ; p ) < ^ ( S L ; p ) (4.19)' From the previous lemma and 4.19 CEH(y H(S L;p))<CE L(y L(S L-p)) (4.20) 4.13, fa  > 0, so gH  > gL. Let &g = gH-  h and Aw = Wl  — U>H Because S^ is the least cost contract to motivate L to pursue p. the incentive com-patibility constraint must bind, therefore  • "  CEL(q L(S L]p)) = w (4.21) Then CEH(qH(SL]p))<w (4.22) / / Because is a mean preserving spread of  S ^  *H(SH-,P)<*H{SL-,P)  (4-23) then CEH(yH(SH]p))<CEH(qH(SL]p))<w (4.24) Since *H(S'H;P)<U H(W)  (4.25) SH  is not an equilibrium contract. The minimum cost contract to motivate H  to pursue p is some contract Sh — (wh — wh + t, g^ — <?#) with t defined  so VH$H\P)  = U„{W)  (4.26) Since t is a fixed  wage payment, and SJJ  is a mean preserving spread of  Si C(S H;p)  > C(S H;p)  = C(SL ;p) (4.27) Since this is true for  any e increase in risk aversion this proves the proposition, so <9C(S;P(S)) dj > 0 (4.28) QED Proposition 4.5 The  expected wage cost of  any least cost contract S motivating p is decreasing in p. Proof:  Consider a least cost contract S = (w,g)  that motivates p = P(S),  and a second least cost contract S = (w,  g) that motivates p = P(S),  where p = p — e.10 From 4.13, < 0, so g > g. Let, Ag be the difference  in the expected value of  the bonus payment between S and S: ^9 = 1(1-  f)(g  - 9) + ^[(1 - V 2) - (1 - P2))9 (4.29) Let A w = w-w (4.30) O A Taking contract S is equivalent to taking contract S (from which the agent derives reservation utility U(w))  and paying Aw for  a lottery with expected payout Ag. Let = ^(1 - P2)U(g  -g) + i [ ( l - f)  - (1 ^ f))]U(g)  (4.31) be the agent's expected utility from lottery. Because \I/(S;p) — xl>(S:p) and the agent has negative exponential utility . U(Aw)  = V(Ag)  (4.32) 1 0 Note that a lower p is a more aggressive investment policy, since the risky project will be chosen for  more realizations of  f. Because the utility function  is concave, one can apply Jensen's Inequality to the left hand side of  4.31 to see that U(Ag)>V(Ag)  (4.33) Combining 4.32 and 4.33, it must be that Ag > Aw. Since C(S;p) - C(S;p) = Ag-Aw ' (4.34) > 0 the. expected cost of  the least cost contract is higher to motivate p — p —  e, than p. The result holds for  any e and proves the proposition, so . dC(S;P(S)) dp QED < 0 (4.35) Proposition 4.6 The  optimal cutoff  point p is increasing in the manager's risk aversion, i. e. plH  > plL where > 7 L. Proof:  The firm's  objective function  is maximized at p such that dV(p)  _ dC(S;p)  . (4.36) dp p=p dp p = p V(p)  is a function  of  p only and independent of  the risk aversion implementing the policy. To show that V  (p)  is concave, note that V(p)  = pv0 + ~ P2)vg + ^(l  ~ p)2vb (4.37) = VQ  — pvg — (1 — p)VB  (4.38) d2V(p)  , — Q j j r = ~ ( v 9 - v b ) < 0 • (4.39) Because V(p)  is concave, and we know that the optimal cutoff  point f>  for  any risk averse agent is less than the first  best, p*F B, to prove the proposition it is sufficient  to show that at every point p < p*F B. Consider two agents, type L with risk aversion j L , and type H  with risk aversion = + e. Consider the the pair of  least cost contracts S L = (w L, <7l),S# = (W H,9H)  that motivate investment policy p for  both agents: PL(SL)  = PH(S H)=P  (4.41) Consider the cost of  motivating either agent to reduce the investment cutoff  point to p — p — 8. Let SL = (WL,9L),  = (VJH,9H)  be the pair of  least cost contracts that motivate the type L and type H  agent, respectively, to pursue the new investment policy. By definition,  these contracts satisfy PL(SL)  = PH(SH)=P  (4.42) Define  AGL and AGH  as the increase in the expected bonus payment under each of  the two pairs contracts: A G L = \{1-P 2){9L-9L)  + \[{1-P 2)-{1  ~ P2)}H  (4.43) A 9H  = ^ ( 1 - f)(g H  - gH)  + k l - P1) - (1 - P2)Y9H  (4.44) Let AW L and A WH  be the difference  in base pay under each of  the two pairs of contracts: A wL = wL-wL (4.45) Aw H  = wH  - wH  (4.46) Let ACI  and ACH  be the difference  in cost for  each of  the two pairs of  contracts: AC l = AgL-AwL (4.47) ACh = AgH-AwH  (4.48) From 4.13, ^ > 0 and < 0, so 9L < 9H  (4.49) 9L < 9H  ' (4.50) (9L~9L)  < (9H~9H)  (4.51) AgL < AgH  (4.52) For agent if,  accepting S H and pursuing investment policy p is the same as accepting pursuing p, and paying A w H for  a lottery with expected payout Ag H . Because SH,  SH  are minimum cost contracts, both satisfy  type Ws reservation utility exactly: *h(SH-,p) = 1>h(SH,P) (4.53) Then it must be that ^H(AG H)  - U H{AW H)  (4.54) . From concavity of  the utility function,  for  this equation to hold, A g H  > AwH,  and therefore A C H  = A g H  - AwH  (4.55) > 0 Define  Aw L as the amount agent L is willing to pay for  a lottery paying the distri-bution A gH\ ^L{Ag H)  =.U l(Aw l) (4.56) Because type Ws utility function  is a concave transformation  of  type Us, type L assigns a higher valuation to the uncertain payment, so Aw L > AwH  (4.57) Define  ACi as AC L - A g H  - AW l • (4.58) Because ACh — AgH — A w H ACl<ACh (4.59) Now consider the reduction in base wage the type L agent takes moving from  SL to S/.: tyL{Ag L) = U L(AW L) (4.60) From equations 4.56 and 4.60, and by concavity of  the utility function,  it must be that • AW L - AW l < AgH  - AgL because the certainty equivalent amount type L is willing to surrender in moving from lottery A g H  (the left  hand side of  4.56) to lottery Ag L (the left  hand side of  4.60) is less than the difference  in expected payout of  the two lotteries. Therefore A C L < A C L < ACH This holds V p > pF B,e,5, therefore  < 0. This is sufficient  to prove the proposition. Because this holds at any point p > p*F B, .the slope of d C { S £ { S ) ) becomes steeper at any point p as 7 increases. Because V(p)  is concave, the point at which dVtp)  dC(S-p)  • • • • /177  7-, p = p = p=p 1 S m c r e a s m S m 7- QED Proposition" 4.7 The  equilibrium expected profit  Tr(S]p))  is decreasing in 7. Proof:  Consider two agents, L with risk aversion coefficient  7 L , H  with risk aver-sion coefficient  jH  = 7 L + e. Let Pi,  i = {L,  H}  be the equilibrium profit  maximizing investment policy for  agent i. From the previous proposition, pL < pH.  Consider the profit  from motivating either agent to pursue investment policy pn• Let SH  t»e the contract that satisfies  pH  = PH(SH),  and define  S^ as the contract which satisfies PH  — PL(SL)-  Then the expected profit  from each of  the contractual relationships to motivate the investment policy is pn given by 7r(S H-pH)  = V(PH)-C(S h;PH)  ' (4.61) •K(S L]pH)  = V(p H)-C(S L]pH)  (4.62) Because the expected cost of  the wage contract motivating any investment policy p is increasing in risk aversion, C(S h;PH)  > C(S L-,PH)  (4.63) Therefore i r (S L - p H )> i r {S H - , p H ) (4.64) The profit  maximizing contract for  the type L agent is S^ and satisfies  pi = PL(SL). By definition TT(Sl;Pl) >TT(S l;ph) (4.65) Therefore . *{SL\PL) > t t ( S h ; P h ) (4.66) This holds for  any e, proving the proposition. QED • • • 4.2.5 Discussion The solution to the single agent case demonstrates that the agent's risk aversion is important from the perspective of  the firm's  shareholders. It also demonstrates that low risk aversion agents are desirable for  two reasons. The first  is the wage cost effect.  The cost of  motivating a given investment policy is increasing in the agent's risk aversion. This occurs because correct effort  and project selection choices can only be motivated by risky pay. The cost of  this is the deviation from optimal risk sharing. Risk averse agents look at a contract and demand that the expected utility it offers  meet their level of  reservation utility. Firm shareholders are risk neutral, and are only interested in the expected cost of  the wage package. The more risk averse the manager, the more costly it becomes for  the firm to offer  a risky pay package meeting a given level of  expected utility. The second reason that low risk aversion agents are more desirable is the project selection effect.  As the agent's risk aversion increases, the best investment policy that the agent can profitably  be persuaded to follow  is increasingly distorted from the first best case. This distortion takes the form of  underinvestment, meaning that risky projects are rejected that the shareholders would ideally prefer  a perfectly  aligned manager to pursue. This occurs because as the agent's risk aversion increases, his risk preferences  are increasingly different  from those of  risk neutral shareholders. 4.3 Model: Competitive Labour Market, Two Agent Types Consider the same problem, but in the context of  a labour market where the shareholders of  n identical firms  must compete with each other for  the services of managers. These managers are identical in every respect, with the exception of  their risk aversion. There are m agents of  type L with low risk aversion 7 L , while the remaining agents have risk aversion 1H  > 1L- M  1S l e s s than n, so all M  type L agents are employed. The remaining n — m jobs are filled  by agents of  type H. An agent's type is private information,  and cannot be communicated to firm shareholders. As such, firms  offer  incentive compatible contracts [ S l , S # ] that leads agents to truthfully  identify  their type through the contract they choose. However, firms  do not operate in isolation. They operate in a competitive labour market, where all firms  will bid for  the services of  desirable low risk aversion (type L) managers. The equilibrium concept is a Nash equilibrium in contract offers. A Nash equilibrium in this market takes the following  form:  m firms  offer  contract S l and hire a type L agent, n — m firms  offer  contract S# and hire a type H  agent, and no firm has an incentive to deviate by offering  some other contract S'. One condition of  such an equilibrium is that t t ( S l ; P l ) = n(S H;pH) 4.3.1 Full information  case To illustrate the importance screening plays in the equilibrium, first  consider the equilibrium in a case where agents' type is observable. Each firm can offer  a contract designed for  a given type of  agent. Both agents demand a certainty equivalent wage of  w to participate in the game. The equilibrium investment cutoff  point, pl, is the same for  each agent as it would be in the single agent case. The investment cutoff  is the one which satisfies dViP ) = dC{  S;p) ( 4 6 7 ) dp p=pi,i dP p=pi,i Contracts are of  the form Sj = (Wi,gi), and are the minimum cost contract that motivate the investment cutoff  Pi. The propositions proved in the previous section show that in equilibrium, type L agents pursue a more aggressive investment policy, PL < PH-  When each agent is held at his reservation utility, it is more profitable  to contract with type L agents. If  there were more type L agents than firms  in the market (M  > n), then there would be no role for  type H  agents. Since ir(S L-,pL) > 7r(SH',PH) when the incentive compatibility constraint binds for  both types, hiring a type L agent is more profitable,  and all firms  would do so. However, when there is a shortage of  type L agents relative to the total number of firms  needing agents (m < n, as is assumed to be the case), type H  agents have a role to play in the labour market. If  firms  were able to contract either type of  agent at their reservation level of  utility; then it would be profitable  to choose type L agents. Any firm contracting with type H  would have an incentive to deviate from such an equilibrium offering  a contract S l = (wl + 6,  gL). This deviation increases type L's expected utility by increasing the fixed  payment by S, while leaving the bonus gi, and therefore  the investment policy pi, unchanged. Because in equilibrium firms  must not have incentive to deviate, the pair of  con-tracts SL ,SH  that would each be optimal in a single agent framework  cannot consti-tute an equilibrium. Firms contracting with type L agents offer  an increase t to the fixed  wage, defined  as: t = n (S L ;p L ) - 7r H (S H ; p H ) (4.68) where Sl is the least cost contract motivating type L to follow  pi in the single agent case, and S// motivates type H  to follow  pH. The equilibrium contract menu is then ^ = (w L + t,gL) (4-69) S H =  (w h,9h) These contracts motivate PL,PH  respectively, and , ' 7r(S L;pL) = n(S H;pH)  (4.70) Type H  agents are kept strictly at their reservation level of  utility.11 Since type L agents create more asset value and are in scarce supply, they earn rents in equilibrium: ^ l \ P l ) > ^ l \ P l ) = U l{W) (4.71) In this full  information  case, there is no need to worry about self-selection  con-straints on the pair of  contracts. The rents paid to type L come in the form of increased base pay, t. The level of  investment under the contract for  type L, pi, is the same as it would be if  only type L agents were present' in the labour market, and the same holds true for  investment under the contract for  H. u I n a competitive market, this reservation utility U  = UH{W)  is such that there is zero expected profit  net of  wages. It is instructive to consider the nature of  rents paid to type L in this case. These rents are not informational  rents, since this is a full  information  case. Rather, these are Ricardian rents accruing to type L because of  their value (from their ability to generate a higher expected firm value) and their scarcity in the market. 4.3.2 Private information  case The game changes when the agent's type is private information.  In equilibrium, firms  offer  screening contracts that elicit truthful  revelation of  type. As such, the contract for  type i must not only satisfy  the constraints of  the single agent case, but also self-selection  constraints. These prevent one type from mimicking the other and choosing the contract designed for  the other agent. The expected profit  for  a firm contracting with an agent of  type i using contract Si = (wi,gi) to motivate pi — Pj(Sj) is n(S l]pi)) = V(p i)-C(S l]pi) (4.72) where V(pi)  = PiVo  + (1 - p])vg + (1 - Pi)2vb (4.73) • C(S, ; P l ) = ^ + (1 -p2)9l (4.74) Define  agent i's expected utility from pursuing investment policy p on contract S as ^(S;p) = PUi(w)  + p2)Ui(w  + g-c) + ^(  1 - p)2U t{w  - c) Then in equilibrium contracts Sj = {S ^ , S /-/} must satisfy (4.75) (4.76) vSirfi)  > MS j-,Pi = Pi{  s , ) ) (4.77) with [i,j]  = [L,H],iy£  j. 4.75 is the incentive compatibility constraint, and assumes that each agent has the same reservation certainty equivalent wage, w. Equation 4.76 defines  a type i agent's individually rational investment cutoff  decision for  a given contract. Equation 4.77 is a self-selection  constraint which requires that agent i prefer  con-tract Sj, that intended for  his own type, rather than the contract intended for  the other agent. ^(S. , ;^ = P i(S :1)) represents the highest expected utility an agent of type i could obtain by taking the contract intended for  the other agent type. Each firm maximizes its expected value, subject to the contracts offered  by other firms  in the competitive market for  managerial labour. In equilibrium, firms  must have no incentive to deviate, and must therefore  be indifferent  as to whether they contract with type L or type H  agents. Therefore v r ( S L , V L )  = T T ( S  H ' , P H ) (4.78) in equilibrium. This equation holds that expected asset value less expected wage costs must be the same for  either agent type in equilibrium. Before  moving to the formal  characterization of  the equilibrium, it is helpful  to discuss the possible outcomes heuristically. First, type H  agents receive the same contract as they would in the single agent case. Because type H agents add less firm value than type L agents, firms  do not have an incentive to bid for  their services beyond the basic level of  utility they would receive in the single agent case. Since firms  have no incentive to change the contract they offer  to type H  agents, is the same contract as would be offered  without the introduction of  a second (more desirable) type of  agent. Type L agents will receive rents, as demonstrated in the full  information  case. What may change, depending on the parameters, is the way in which those rents are paid. Relative to the single agent type case, type L agents receive rents based on their ability to add more firm value than type H. Because in equilibrium firms must be indifferent  between hiring either type of  agent, and type L agents create higher valued firms  gross of  expected wage costs, expected wage costs, and ultimately expected utility, are higher for  type L. However, the manner in which type L agents receive their rents depends on the relative value type H  agents place on their own contract, SH,  and the contract in-tended for  type L, S^. As long as type H  agents prefer  their own contract, firms  can pay rents to type L agents in the most efficient  way possible: increase base pay on S/,. If  there is some level of  increased base pay, without changing type L bonus pay, at which firms  are indifferent  between agent types, then type L agents receive rents in the form of  increased base pay only, and there is no change in investment policy. However, if  base pay to type L increases to the point that type H  agents would start to prefer  the type L contract if  base pay increases further,  and firms  still prefer type L agents, then rents paid to type L agents must take a second form:  increased bonus pay, in addition to the increased base pay. This solution has the advantage that the more risk averse type H  agents find  the bonus pay less attractive than do type L agents, thereby performing  the screening function  required of  the contract menu. However, increased bonus pay changes type L investment policy from the single agent case, and is an inefficient  means of  paying rents relative to the full  infromation solution. Because the rents come in the form of  risky pay, risk averse agents value the bonus payments less than they do certain payments. Properties of  the equilibrium Lemma 4.2 The  equilibrium wage contract designed for  the type H  agent, SH,  is the same contract that would be offered  in a labour market populated by type H  agents only. Proof:  As in the single agent case, the incentive compatibility constraint (4.75) must bind for  the type H  agent. Because it is more costly to motivate type H  agents to pursue any investment policy p, there is no reason for  firms  to work to attract a type H  agent over a type L agent. As such, firms  contracting with type H  agents design the contract which maximizes expected profit.  This is the investment cutoff PH  that satisfies MP> = 0C(S„; P) dp p,j>„ dp p,iH These are exactly the same equilibrium conditions as in the single agent case, so SH, is unaffected  by the presence of  the type L agent in a labour market setting. QED Proposition 4.8 Type  L agents earn rents in equilibrium, and the incentive com-patibility constraint (equation  4.75)  does not bind for  type L agents. Proof:  Consider a contradiction. Let S^ be the equilibrium wage contract, in-tended for  type L agents, offered  by m firms.  Let SH  be the contract offered  by n — m firms,  intended for  type H  agents. If  the type L agent's incentive compatibility constraint binds, then Vl(SL,PL) = UL{w) (4.80) From the previous proposition, we know that type H 7s  incentive compatibility con-straint binds: ^h{Sh\PH) = UH(w) (4.81) From the previous section, the type L agent can generate a higher certainty equiv-alent utility than type H  on any contract. Therefore CEL(^ L(S H-,PL  = PL{SH)))>CE H{* H(S H-P„))  = W  (4.82) > CE l(^ l(S l-PL))  (4.83) *L{SH\PL)  > ¥L(SL\PL)  - (4.84) Equations 4.82 - 4.84 show that SL violates the self-selection  constraint 4.77. Therefore,  type L's incentive compatibility constraint cannot bind in equilibrium, and type L agents earn rents. QED Proposition 4.9 If  for  an equilibrium pair of  contracts SL,SH  the self-selection constraint Jf.,11  does not bind for  type H  agents, then SL motivates the same invest-ment policy pi as would be optimal in a labour -market populated by type L agents only. Proof:  Let Sn be the equilibrium contract offered  to type H  agents. Let S^ = (WL,9L)  be the contract that motivates the same investment policy, pL, as would be optimal in the single-agent case populated only by type L agents, and satisfies  the self-selection  constraints so that > ^(ShSPL  = PL(S H))  (4.85) > ^h(S l;PH  = PH(S l)) (4.86) as well as the equilibrium profit  condition n(S L;pL) =ir(S H-,pH)  (4.87) Since SL motivates the single agent case investment policy, DV<P ) _dC(S H-P) dp p=pl,L dp p=f )^L Let Si be a contract that motivates a different  investment policy, pL, and keeps the type L agent's utility unchanged: *L{SL;PL)  = *L{SL-3L)  (4-89) where S L = ( w L - m , g L + . S ) (4.90) Because Pl{Sl)  - P l ( S l ) , any 8 implies lower expected profit  for  the firm,  since 9V(P ) f dC(S L-p) ( 4  9 1 ) dp p=pL,L dp p=pL,L Because this is true for  all 8, there exists no contract which is a Pareto improvement over Si, which proves the proposition. QED Proposition 4.10 Let PL be the level of  investment motivated by optimal contract for  a labour market populated by type L agents only, and let CJL  = Gi(pi)  be the bonus payment that motivates the type L agent to pursue policy p^. Let SL,SH  be the equilibrium contracts. If  there exists a contract SL — (WL,9L)  f or which (i)  the type H  agent's  self-selection  constraint binds, and (ii)  for  which TT(SL',PL)  > 7r(SH',PH) then the equilibrium contract for  type L motivates a more aggressive investment policy than in the single agent case. Proof:  SL,SH  cannot be an equilibrium as 7r(SL',PL)  > K(SH',PH)-  The equilib-rium type L contract, S^, must satisfy  H(SL',PL)  — ^(SHIPH)-  Therefore  7r(S/_;,PL)  < n{S L;pL). Because *H(S h;PH)  = ^H((S l-,PH  = PH(SL))  (4.92) then for  any contract S^ = (UIL  + E, 9L) it will be the case that • VH(SH;PH)  <*H(&L\PH)  (4-93) \ as the base wage is higher for  Si but the bonus payment is the same for  both contracts. This makes the agent strictly better off,  and violates the self-selection  constraint 4.77 for  agent H. Then the equilibrium contract S^ must be such that w^ < WL-The equilibrium contract menu S ^ . S// must therefore  satisify *L(SL\PL)  > *L(SL;PL)  - (4.94) *H(SH;PH)  < VH(SL\PH  = PH{SL))  (4.95) tt(S L-pL) = TT(S H ]pH)  (4.96) For any contract SL = (w L — e,gL), e > 0, ^ L ( S l ; P L ) < ^ L ( S l ; p l ) (4.97) Because WL  < WL,  by equation 4.94: 9L > 9L (4.98) Because gL > gL, and ^ < 0, pL > pL. Finally, for  there to exist an equilibrium contract S^, it must satisfy ^H{SL',PH  = PH(SL))  = *H{SL\PH  = PH.{S L)) (4.99) > V l (S l ;P l ) (4.100) Let AwL = wL — wL, and Ag L = gL — gL. SL is equivalent to S^ plus paying Awi for  a lottery paying Ag L in.the event the good state is realized. From 4.99, it must be that CEH(V H(Ag L)-AwL = 0 (4.101) Because type L can generate higher certainty equivalent utility from any lottery CEL(V L(Ag L)-AwL>0 ' (4.102) Therefore  a contract satisfying  equations 4.94 to 4.96, 4.99 and 4.100 can be found. This proves the proposition. QED Corollary 4.2 When  equation J^.li  binds for  type H  agents, the equilibrium con-tract menu may motivate type L agents to pursue a policy of  overinvestment relative to the first  best case. This is a natural extension of  the previous proposition. A parametric example is sufficient  to demonstrate the existence of  such equilibria. Consider an example where {v B = 0,v0 = 100, VQ  — 400, c = 2 ,7 L = 0 . 1 , = 0.5} and both agents demand a utility equal to that provided by a certainty equivalent wage of  w = 5. The optimal contract to be offered  to the type H  agent is SH  = {WH,9H)  = (4.606,10.179). This maximizes expected profit  TR(SH',PH  — PH(SH)  = 0.636) = 175.058. The equilibrium contract S^ that satisfies  both the type H  self-selection  con-straint and the equilibrium labour market requirement that 7r(SL',PL)  — TT(SH',PH)  IS = (4.593,65.995). Type H's  self-selection  constraint is satisfied;  ^H{SH',PH) ^h{S l ;Ph = Pf/(S L) = 0.632) - -0.082. The expected profit  when the type L agent takes the contract is 7r(SL',PL  = PL(SL)  = 0.182) = 175.058, the same as the expected profit  when contracting with a type H  agent on S In a full-information  environment with observable effort,  the first-best  level of investment is given by p*F B = = 0.245. Comparing this cutoff  point to the equilibrium cutoff  point of  PL  = 0.182, it is clear that in this equilibrium type L agents are provided incentives to overinvest relative to the first-best  case. Proposition 4.10 In  equilibrium, firms  contracting with type L agents have higher ex ante variance of  expected firm  value than do firms  contracting with type H  agents. Proof:  The ex ante distribution of  firm value, after  the agent is hired but before the updated signal f  is received, is trinomial. When the investment cutoff  is p expected firm value is given E(V)=pv 0 + (l-p 2)vg + {l-p) 2vb • (4.103) The variance of  firm value is given Var{V)  = E{V 2) - [E{V)} 2 (4.104) whose first  derivative with respect to p is given [-  (1  - p)v2b - pvI  + vl) - [(-  •1 + p)vb -pvg + vo] [(1 - pfv h + vg- p\ + 2pv0} (4.105) This expression is negative for  all p £ (0,1) where vb < VQ  < vg. Because PL < PH, the ex ante variance of  firm value is greater for  firms  hiring type L agents. QED Discussion There are two key changes in the case where multiple firms  compete for  the services of  two different  types of  agents. The first  is that firms  introduce a menu of  contracts, and these contracts perform a screening function  to distinguish between agent types. The second is that because firms  have an incentive to bid against each other for  the scarce services of  type L managers, type L managers capture rents in equilibrium. The nature of  rents in this model differs  from that of  most screening models. Typically, the "good" type in a given model earns rents due to the self-selection constraint. The desirable agent receives rents in order to elicit truthful  revelation of type. A portion of  the rents the desirable type L agent receives in the model comes from this source. Firms cannot offer  the type L agent the same contract they would in a single-agent setting, because the type L contract can do better by accepting the contract intended for  type H. However, this, is not the primary source of  rents to the agent in the model. Rents accrue to type L agents because their lower risk aversion leads them to make less distorted investment choices relative to a first-best  (or risk neutral agents) case. In a labour market where firms  have the ability to bid up the prices of  agents, type L agents capture the extra value that they create. As long as the type H  agents' self-selection  constraint does not bind, increased pay to type L agents takes the form of  increased base pay. This is the most efficient  type of  payment from a risk sharing perspective. The interesting cases are those where the high type's self-selection  constraint is binding. The nature of  the equilibrium is much different  than the result in a single-agent case. Here, firms  contracting with type L agents must design a contract which performs  a screening function,  in that it has to satisfy  type H  agents' self-selection constraint. At the same time, in equilibrium the contract provides rents to type L to the degree that firms  are indifferent  between contracting with either type of  agent. Put another way, type L agents capture all of  the extra value they create. The solution which satisfies  both the screening requirement and the labour market requirement is one where rents to type L take the form of  far  more bonus compensation than they receive in a single agent type setting. The use of  bonus pay takes advantage of  the different  agent types' feelings  about risk. Type L agents value the bonus compensation more highly and earn rents by taking the contract. Type H  agents assign a much lower value to this riskier contract,, and as a result their self-selection constraint is satisfied. The interesting result that emerges from this labour market is that the presence of  type H  agents can cause firms  to increase incentive pay to type L agents beyond what they would receive in a single-agent market. The contracts for  type L agents can exhibit reduced project selection distortion relative to the single agent case. In-creasing risky pay leads to a more aggressive investment policy, reducing the degree of underinvestment that arises in the single-agent case. However, this comes at the cost of  increased wages and poorer risk sharing than in the single-agent case. Clearly, the wage cost and risk sharing effects  must outweigh the'value of  the improved project selection, otherwise the level of  investment motivated in the dual-agent case would also have been optimal for  the single-agent case. The most interesting equilibrium is in cases where type L agents actually overin-vest relative to the first  best case. This provides a stark example of  just how important the contracts' screening role can be in the two agent case. Risky pay, expensive due to the deviation from optimal risk sharing that is required to make it work, is used to the point that type L agents choose to overinvest. The need for  screening in these cases causes firms  to offer  a risk averse agent a contract of  higher power than they would offer  even a risk neutral agent, with whom risk sharing concerns would be irrelevant. The intuition for  this relates to the reason that type L agents earn rents in the model. Rather than being paid rents to ensure that they don't mimic type H, the rents paid to type L have more to do with a traditional labour supply and labour demand curves. There are relatively few  type L agents, they add more value, and in equilibrium they are paid more to reflect  this. This causes equilibria where the type H  agent's self-selection  constraint binds instead. This is the reason that rents paid to type L take a form (bonus pay) that is inefficient  from the. perspective of  straight risk sharing-project selection tradeoff  concerns. The bonus is not valued by type L agents anywhere near its true value. In cases where overinvestment results, some of the "rents" are paid in the form of  asset value destroyed by overinvestment. This is clearly not desirable, but it is a consequence of  firms'  efforts  to avoid value being lost because type H  agents choose to underinvest in equilibrium. Finally, it is informative  to compare the interaction between firm risk and risk sharing in this model with that of  many traditional principal-agent models. Generally, in models where firms  provide agents with risky pay in order to motivate effort,  the amount of  risky pay they offer  is decreasing in the firm's  (generally exogenously specified)  firm risk. The expected cost of  providing risk averse agents with incentive pay to provide them a given level of  expected utility is increasing in firm risk. As the firm's  returns become riskier, the tradeoff  between the effort  incentives from risky pay and the cost of  the risky pay causes the firm to choose lower powered contracts. This contrasts with the empirical literature, where Prendergast finds  that the link between firm risk and the level of  risky pay provided is quite mixed. In this model, firm risk is not an exogenous parameter around which firms  make contracting choices. Rather, firm  risk is an endogenous outcome of  the contracting choices a given firm  makes. In this model, agents influence  firm risk because they have the discretion to make project selection decisions. Firms that contract with type L agents provide greater incentive pay than do firms  that contract with type H agents. The higher power of  type L contracts, combined with type L agents' lower risk aversion and project selection ability, leads firms  managed by L-type agents to have higher ex ante variance in firm  value than firms  managed by type H  agents. This is the exact opposite predicted by a model where managers do not influence  project selection and firm risk is exogenous. 4.4 Conclusion The paper develops a model where incentive contracts are designed to elicit ef-fort,  motivate properly aligned project selection and investment decisions, and screen potential candidates. I demonstrate the importance of  screening in a setting where agents of  differing  risk aversion populate the labour market. Firms have an incentive to bid for  low risk aversion agents, who can be encouraged to pursue a less distorted investment policy. The resulting labour market equilibrium leads to outcomes where contracts have much greater power than they do in versions of  the same model where screening is not a consideration. This result sheds new light on why very high power contracts are often  observed empirically. Such contracts are very hard to justify  using a traditional model which trades off  costly risk sharing against the need to motivate effort.  The introduction of a screening component to the model provides a reasonable justification. This paper also incorporates the realistic feature  that high-level managers do not only exert productive effort  on a given project. They also crucially make investment and project selection decisions that affect  the distribution of  firm returns. In such a setting, managerial risk aversion becomes an extremely important consideration in contract design, since managers with different  degrees of  risk aversion make different investment decisions when faced  with the same contract. The results provide new understanding of  the roles played by incentive compen-sation, as well as the optimal degree of  incentive power provided by such contracts. 4.5 Bibliography [1] Carpenter, Jennifer  N., "Does Option Compensation Increase Managerial Risk Appetite?", Journal  of  Finance,  October 2000, V.55 No.5, 2311-2331. [2] Core, John E., Wayne Guay and David F. Larcker, "Executive Equity Compen-sation and Incentives: A Survey", 2002, working paper. [3] Core, John E. and Jun Qian, "Project Selection, Production, Uncertainty, and Incentives", 2002, working paper. [4] Grossman, Sanford  J. and Oliver D. Hart, "An Analysis of  the Principal-Agent Problem", Econometrica, January 1983, V.51 No.l, 7-46. [5] Hall, Brian and Kevin J. Murphy, "Stock Options for  Undiversified  Executives", 2001, USC working paper 01-16. [6] Harris, Milton and Artur Raviv, "Some Results on Incentive Contracts with Appli-cations to Education and Employment, Health Insurance, and Law Enforcement", American Economic Review, March 1978, V.68 No. 1, 20-30. [7] Haubrich, Joseph G., "Risk Aversion, Performance  Pay, and the Principal-Agent Problem", Journal  of  Political Economy, April 1994, V.102 No. 2, 258-276. • [8] Holmstrom, Bengt, "Moral Hazard and Observability", Bell Journal of  Economics, 1979, V. 10 No. 1, 74-91. [9] Holmstrom, Bengt, "Managerial Incentive Problems: A Dynamic Perspective", Review of  Economic Studies, 1982, V.66 No. 1, 169-182. [10] Holmstrom, Bengt and Paul Milgrom, "Aggregation and Linearity in the Provi-• sion of  Intertemporal Incentives", Econometrica, March 1987, V.55 No. 2, 303-328. [11] Jensen, Michael C. and William Meckling, "Theory of  the Firm: Managerial Be-havior, Agency Costs and Ownership Structure", Journal  of  Financial  Economics, October 1976, V.3 No. 4, 305-360. [12] Jensen, Michael C. and Kevin J. Murphy, "Performance  Pay and Top-Management Incentives" , Journal  of  Political Economy, April 1990, V.98, 225-264. [13] Jullien, Bruno, Bernard Salanie and Francois Salanie, "Screening Risk-Averse Agents Under Moral Hazard", 2000, working paper. [14] Lambert, Richard. A., "Executive Effort  and Selection of  Risky Projects", RAND Journal  of  Economics, Spring 1986, V.17 No. 1, 77-88. [15] Lewellen, Katharina, "Financing Decisions When Managers Are Risk Averse", 2003, MIT working paper 4438-03. [16] Murphy, Kevin J., "Executive Compensation", 1999, working paper. [17] Nohel, Tom and Steven Todd, "Stock Options and Managerial Incentives to Invest", 2001, working paper. [18] Prendergast, Canice, "The Tenuous Trade-off  between Risk and Incentives", Journal  of  Political Economy, 2002, V.110 No. 5, 1071-1102. [19] Ross, Stephen A., "The Economic Theory of  Agency: The Principal's Problem", American Economic Review, May 1973, V.63 No. 2, 134-139. [20] Ross, Stephen A., "Compensation, Incentives, and the Duality of  Risk Aversion and Riskiness", Journal  of  Finance,  Feb. 2004, V.59 No. 1, 207-225. [21] Rothschild, Michael and Joseph E. Stiglitz, "Increasing Risk: I. A Definition", Journal  of  Economic Theory,  1970, V.2 No. 3, 225-243. [22] Serfes,  Konstantinos, "Endogenous Matching in a Market with Heterogeneous Principals and Agents", 2005, working paper. [23] Wright, Donald J., "The Risk and Incentives Trade-off  in the Presence of  Het-erogeneous Managers", Journal  of  Economics, 2004, V.83 No. 2, 209-223. Chapter 5 Conclusion In the first  essay I seek to explain how tort liability affects  a firm's  optimal capital structure. While other papers have made the point that limited liability will affect economic agents' incentives with respect to tort risk, very few have sought to endog-enize the firm's  decision about in which states it will be solvent. A key characteristic of  tort risk is that its impact on cash flows  available to security holders depends on the structure of  security holders' claims. Put another way, capital structure matters greatly when determining the potential expense payable to tort claimants. Recogniz-ing this, firms  with exposure to tort liability will have an incentive to adjust capital structure to respond optimally. The lower creditor priority of  tort claimants implies two effects  when debt and tort risk interact. The first  is that tort liability brings about an increased probability of  bankruptcy. Where this effect  predominates, the firm will choose to move away from debt. The second effect  is that debt provides an asset shielding advantage, preserving cash-flow  rights for  the firm's  debt holders at the expense of  tort claimants. Where this effect  is dominant, increased tort risk will cause the firm to choose more debt. I specify  two simple models to examine the interaction of  these effects,  one where firm returns are distributed continuously over an interval, and another where firm returns are distributed binomially. The different  results from these two illustrations demonstrate the importance of  assumptions regarding firm cash flows.  Depending on the nature of  the firm's  returns, and the values of  the various input parameters, either the bankruptcy effect  or asset shielding effect  can dominate. Tort liability is a major source of  risk for  firms  today. I have shown why it is unique, and why firms  must consider its unique properties when determining the optimal capital structure. Empirical work studying how firms  do adjust their capital structure to address changes in tort risk is a potentially fruitful  avenue for  future research. In the second essay we explore the capital structure of  insurers. The focus  is on the impact of  aggregate uncertainty, or dependence among risks, since this is the source of  an insurer's incentive to issue equity. Insurance firms  respond to the shocks of  increased risks by taking all or some of  the following  actions: placing limits on the number or coverage of  contracts that they offer;  raising premium for  the policies that they issue; and raising more equity. We analyze the equilibrium mixture of  these responses in a competitive insurance market, and find  that the impact of increasing uncertainty on the equity decision depends on the nature of  aggregate uncertainty. Where this uncertainty is in the size of  losses, equity increases with uncertainty; where the risk dependence is in the events of  losses, equity first  increases then decreases with uncertainty, providing that individuals are not too risk averse. The latter result follows  from a tradeoff  between two effects,  which we label the input effect  of  uncertainty, and the output effect.  In both cases, however, the ratio of  equity to insurance revenue increases. We extend the model to look at the effect  of  a cost difference  between internal equity (less costly) and external equity (more costly). This extension leads to the hypothesis that firms  with greater internal equity will tend to use less leverage. We test both hypotheses directly on a sample of  852 U.S. property and casualty i stock insurers over a sample period from 1999-2004. We find  support for  both of our hypotheses.. Firms that have higher variance in their loss ratio, our proxy for uncertainty, use significantly  less leverage, supporting our theory that uncertainty and leverage are negatively correlated. Firms that have been recently profitable, implying greater internal capital, use significantly  less leverage. This supports the theory that there is a cost advantage to internal over external equity, which is at the core of  recent theories of  insurance market dynamics. In the third essay, I develop a model where incentive contracts are designed to elicit effort,  motivate properly aligned project selection and investment decisions, and screen potential candidates. I demonstrate the importance of  screening in a setting where agents of  differing  risk aversion populate the labour market. Firms have an incentive to bid for  low risk aversion agents, who can be encouraged to pursue a less distorted investment policy. The resulting labour market equilibrium leads to outcomes where contracts have much greater power than they do in versions of  the same model where screening is not a consideration. This result sheds new light on why very high power contracts are often  observed empirically. Such contracts are very hard to justify  using a traditional model which trades off  costly risk sharing against the need to motivate effort.  The introduction of a screening component to the model provides a reasonable justification. I also incorporate the realistic feature  that high-level managers do not only exert productive effort  on a given project. They also make investment and project selection decisions that affect  the distribution of  firm returns. In such a setting, managerial risk aversion becomes an extremely important consideration in contract design, since managers with different  degrees of  risk aversion make different  investment decisions when faced  with the same contract. The results provide new understanding of  the roles played by incentive compensation, as well as the optimal degree of  incentive power provided by such contracts. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0100579/manifest

Comment

Related Items