Destructive Agents, Finance Firms andSystemic Risk

N. Bilkic and T. Gries

February 7, 2014

Abstract

Popular opinion suggests that malfunctioning, poorly designed incen-tive schemes in �nancial �rms that encouraged greed and involved exces-sive salaries were responsible for the excessive risk taking that eventuallyled to the 2008 �nancial crash. In this paper we discuss this claim in atheoretical model. We use a modi�ed version of delegated portfolio choiceapproach with performance contracts. If, in this modi�ed model, we al-low for the existence of destructive agents - when maximizing their privateutility - each �nancial �rm will take excessive risks. As a result the �nancesector develops systemic risk. We de�ne systemic risk as ine¢ cient andexcessive risk that is chosen in an endogenous and stable manner by theaggregate market.

JEL classi�cations: D82, D86, G14Keywords: delegated portfolio choice, systemic risk,

destructive agent, adverse selection

�) Corresponding author: Thomas Gries: thomas.gries@notes.upb.deEconomics Department (C-I-E), www.C-I-E.orgUniversity of Paderborn, Germany

Co-author: Natasa Bilkic, natasa.bilkic@notes.upb.deEconomics Department (C-I-E), www.C-I-E.orgUniversity of Paderborn, Germany

1

[�rst page without authors]

Destructive Agents, Finance Firms and Systemic Risk

Popular opinion suggests that malfunctioning, poorly designed incentiveschemes in �nancial �rms that encouraged greed and involved excessive salarieswere responsible for the excessive risk taking that eventually led to the 2008�nancial crash. In this paper we discuss this claim in a theoretical model. Weuse a modi�ed version of delegated portfolio choice approach with performancecontracts. If, in this modi�ed model, we allow for the existence of destructiveagents - when maximizing their private utility - each �nancial �rm will takeexcessive risks. As a result the �nance sector develops systemic risk. We de�nesystemic risk as ine¢ cient and excessive risk that is chosen in an endogenousand stable manner by the aggregate market.

JEL classi�cations: D82, D86, G14Keywords: delegated portfolio choice, systemic risk,

destructive agent, adverse selection

2

1 Introduction

In a 2008 article discussing the crash of the �nancial system, the New YorkTimes argued, �Between 1998 and 2008 Rubin was a top o¢ cial at Citygroup,where he received a cumulative $150 million in compensation. His main impacton bank policy was to push for the kind of aggressive risk taking that crashed the�rm. Rubin believed that Citigroup was falling behind rivals like Morgan Stan-ley and Goldman, and he pushed to bulk up the bank�s high-growth �xed-incometrading, including the CDO business ...�.1 Similarly, in 2012 was written in theHarvard Business Review, �... the rise of the alternative assets industry hasaltered behavior through much of the �nancial sector. Financial markets basedcompensation has become the norm in modern American capitalism. Unfortu-nately, the idea of market based compensation is both remarkably alluring anddeeply �awed. The result has been the creation of perhaps the largest and mostpernicious bubble of all: a giant �nancial incentive bubble.�2 Such anecdotalexamples stand for more general popular claims such as �greedy�managers areresponsible for excessive risk taking and caused the 2008 crash that was fol-lowed by one of the largest �nancial crises ever. In other words, in the public�sopinion, the blame for the crash largely lies with malfunctioning and poorly de-signed incentive schemes in the �nancial institutions that encouraged excessiverisk taking by payment incentives.That said, academic research has so far not supported such a clear negative

appraisal of intra-�rm incentive systems. Therefore, in this paper we examinethree central questions raised by these popular claims. (i) Can a principal-agent structure with asymmetric information, moral hazard, and hidden actionin �nancial �rms explain the emergence of the �nancial crisis and "systemicrisk"? More speci�cally, (ii) how do contractual incentives cause excessive risk-taking at individual �rm level? And �nally, (iii) how do intra-�rm incentivesystems cause systemic risk at sector level?The goal of this paper is to understand what happens in terms of risk-

taking behavior in �nancial �rms and the �nance sector if we just apply simpleperformance contract schemes widely used in the �nance industry. The goalis not to determine what may have been the optimal contract to prevent the�nancial crisis; rather, we seek to understand what may have encouraged oreven caused the crisis with respect to an incomplete understanding of incentivesand payment schemes within �nancial �rms.To answer these questions we suggest a modi�ed version of a "delegated

portfolio choice" model. One of the main modi�cations in our model is that weallow for the existence of agents who deliberately generate wrong informationfor their own advantage and hence are not only disloyal but even act deliberatelyharmfully when maximizing their private utility. In analogy to Baumol (1990)we refer to these agents as "destructive agents".3 If such destructive agents

1New York Times. 2008, Nov/23, The Reckoning: Citigroup pays for a rush to risk.2M. Desai, Harvard Business Review, p.124, March, 2012.3At this point we import the concept from entrepreneurship literature. The notion of a

"destructive entrepreneur" Baumol (1990) transfer to managers being a destructive agent.

3

manage their �rms�portfolios, each �nancial �rm will take excessive risks andthe �nance industry will develop systemic risk. In other words, we want to �ndout if excessive risk (that is, more than e¢ cient risk) is systematically takenwhen a simple performance contract is the dominant compensation scheme, ascan be observed in the �nancial industry. Wrong kinds of incentives encouragedeliberately destructive agents to generate risk externalities and systematicallyallocate assets against the principal�s interest.Our point of departure is the delegated portfolio choice model. In the one-

period setting, the broader discussion starts with the contribution by Bhat-tacharya and P�eiderer (1985), who introduced a model with asymmetric infor-mation on a risky asset. A better informed agent needs to be motivated to usethis information in the principal�s best interest. Hence, a performance relatedcontract is suggested.In the debate that followed, the design of the optimal contract became a

major point of discussion. In a classic setting an agent has to invest e¤ort intoacquiring information about the risky asset. If the agent�s e¤ort to obtain thisinformation is observable and there is no asymmetry of information, choices arejointly determined as they are for one common unit and a �rst-best solution isderived. In this case Stoughton (1993) shows that if both, agent and principal,are risk-averse a contract simply implies optimal risk-sharing between agent andprincipal.The more interesting case, however, remains the one with information asym-

metry. While information asymmetry and associated moral hazard problems donot allow for a �rst- best outcome under linear sharing rules, models such asthat of Holmstrom and Milgrom (1987) show conditions under which linearcontracts can be best in a second-best world. However, in the delegated port-folio choice context Stoughton (1993) and Admati and P�eiderer (1997) bringup another important phenomenon, namely the irrelevance result. That is,in standard principal-agent models the payment incentive is related to projectperformance and hence makes the agent work harder on behalf of the princi-pal. Since standard modeling in delegated portfolio models with linear contractsallow agents not just to invest e¤ort into obtaining information but also into fur-ther controlling the portfolio outcome, the principal�s incentive instruments canbe neutralized and not directly used to motivate the agent. Hence the simplerelationship known from standard principal-agent models no longer works in thedelegated portfolio approach, as it is modeled in the literature. Stracca (2006)even concludes in a survey that "more fundamentally, a compensation contractwhich is optimal (from a second best perspective) in a general class of delegatedportfolio management problems is not known even under the assumptions ofHolmstrom and Milgrom (1987). Generally speaking, the literature has reachedmore negative rather than constructive results, and the search for an optimalcontract has proved to be inconclusive even in the most-simple settings."However, there are some results that can be regarded as major benchmarks

in the debate, even if they are derived within a speci�c setting. Admati andP�eiderer (1997) suggest that a quadratic contract may solve the problem con-nected to the irrelevance result, hence the agent can respond to the signal in

4

a non-linear way. Li and Tiwari (2009) suggest that, for non-linear contractsthat promise a �xed payment, a proportional asset-based fee and a benchmark-linked fulcrum fee it is always optimal to include a benchmark-linked optiontype bonus incentive fee (with the appropriate choice of benchmark).Not only the question of optimal contracts is considered in the literature;

the question of how certain contracts impact managers�decisions and sometimesprincipals�interests is also relevant. With respect to the question of excessiverisk-taking, the results are mixed. The problem of symmetric or asymmetriccontracts is of particular interest. Starks (1987), for instance, compares a sym-metric fulcrum performance fee with an asymmetric bonus contract and arguesthat the symmetric performance fee is preferable because it can at least alignrisk attitudes between agent and principal.Comparing an asymmetric incentive contract with symmetric fulcrum con-

tracts, Das and Sundaram (2002) �nd in a signaling model that incentive feesmay lead to more risky portfolios due to agent self-selection. In this way, asym-metric contracts tend to lure less informed agents into the business.Carpenter (2000) looks at a risk-averse manager compensated with a call-

option contract and obtains opposite e¤ects concerning the risk that a manageris willing to take. Chen and Pennacchi (2002) and Ross (2004) obtain similarresults. Ross identi�es a number of e¤ects depending on assumptions about theutility function and speci�c contract design. E.g., an agent with an option-likecompensation contract may even choose a level of volatility that is lower thanthe one they would take if they traded on their own. Under a value-at-risk (VaR)constraint Sheng et al. (2012) show that a linear performance-based contractcan motivate agents to acquire information. However, a VaR constraint mayincrease the moral hazard problem.Beyond the above discussion, the issue of limited liability has become an

important issue in the debate on risk taking. In a rather general approachGrinblatt and Titman (1989) illustrate that under limited liability, agents tendto take on a riskier portfolio. Palomino and Prat (2003) depart from Gollieret al. (1997) and transfer the limited liability issue to a full modeled principalagent setting.4 Palomino and Prat explicitly mention that the agent can conductsabotage actions, an approach we think is very realistic and hence will becomepart of our own modeling later on. In their modeling they are unable to �nd anoptimal linear contract under limited liability.It is evident that the literature is aware of asymmetric information, moral

hazard, and the principal�s enormous di¢ culties in controlling the agent�s ac-tivities. However, in the setting so far the agent acts as a more or less willinginstrument and just needs to be su¢ ciently motivated to put more e¤ort intorealizing the principal�s interests. Hence the agent is reduced to a very smallset of choices that in no way re�ect the enormous range of unobservable actionsthey could conduct. There are many more dimensions in which they could actunobserved to realize their personal interests than, e.g., the simple choice of

4Also Hellwig (1994) and Biais and Casamatta (1999) study moral hazard with both e¤ortand risk. They do not consider a full portfolio choice setting.

5

e¤ort level in obtaining asset information. Only very few authors mention theseother potential activities as part of the agent�s strategy.5 Since this notion ofthe agent using information asymmetry to develop their own active strategyand actions is not yet included in the models, we use this idea as a point ofdeparture.Once again, let us be clear that we do not intend to determine an optimal

contract within a certain, even more complex, delegated portfolio setting. Thenovelty of this paper is that we explicitly model a destructive agent who exploitsinformation asymmetry to deliberately manipulate information to realize theirown interests and simultaneously harm the principal under performance contractconditions. We also want to discuss how this kind of agent may be the selected,employed and successful type in the market for managers via adverse selection,and may also be responsible for excessive risk taking in individual �nance �rmsand in the �nance sector.In a nutshell, we suggest a very simple and slightly modi�ed version of a

delegated portfolio model. In this model the agent has perfect information (noe¤ort to collect information is needed). The principal can observe only theportfolio return and look at observable indicators, like the expected return ande.g. volatility as risk indicator. However, pure portfolio volatility as observablerisk indicator is not su¢ cient for determining the individual �nancial �rm�s trueportfolio risk performance. The reason is simple. For the �rm�s portfolio as-sessment the principal is interested in the idiosyncratic risk performance of themanaged portfolio. From the principal�s perspective observable volatility coversidiosyncratic risk components and elements which are driven by other system-atic (common) risk conditions. As information asymmetry does not allow theprincipal to separate between idiosyncratic and systematic risk, the principal�sperception and assessment of portfolio performance is subject to interpretations.Hence, the perception of risk is in�uenced by the information the principal ob-tains from all kinds of sources, including from the agent, who is compensatedaccording to portfolio performance. As a result, the agent can use the princi-pal�s imperfect information and their own information advantage to put e¤ortinto manipulating the principal�s perception of idiosyncratic risk for their ownbene�t. The agent chooses a high-return and high-risk portfolio while activelyconcealing and manipulating information about the true idiosyncratic risk (in amanner that best serves their personal objectives). Due to asymmetric informa-tion they appear to perform well and hence will earn a high performance-relatedincome. When principals negotiate such contracts with potential agents we ob-serve an adverse selection in the market for agents. The principal cannot seethe true characteristics of agents (like the ability to manipulate information, orthe degree of risk aversion) and hence selects bad agents. As a result we obtainan ine¢ cient excessive risk-taking in each �rm. What is more, such excessiverisk is systematically taken by the entire market. This endogenously evolvingexcessive risk is identi�ed as "systemic risk."

5E.g., Palomino and Prat (2003) speak of sabotage as a potential action on the part of anagent.

6

2 Finance Firm with Delegated Portfolio Choice

In this section we look at a single �nance �rm that (i) needs to negotiate anddraw up a contract between the principal and the agent that is going to managethe portfolio and (ii) has to select from the market of agents the one that itbelieves will serve its purposes best.

2.1 Describing the Structure of the Firm

The �nance �rm�s business is to manage a given amount of wealth.Wealth is either owned by the principal itself, or the principal is liable forvalues deposited by external depositors. We assume no leverage (no bank-speci�c model) even if it can be included. Pro�ts are attained through returnsfrom a portfolio chosen by the agent. For simplicity, there are no operating ormonitoring costs.

The �rm�s portfolio consists of two kinds of assets, a risk-free asset(government bond) with a certain return RB , and a set of risky capital as-sets characterized by projects with i.i.d returns, an expected return ERK ; andan identical volatility �K representing the true risk. These assumptions arean extreme simpli�cation, since we assume away covariate and systematic riskcomponents for describing risk in this model. We do that because we would liketo discuss the most simple case that can make the point to easily distinguishbetween "true risk" in this model and the perception of risk by the principal.Returns of risky assets are normally distributed and the portfolio share of thisasset is b. Hence, the expected return of the portfolio and the true portfolio riskis

ER = (1� b)RB + bERK (1)

�2 = b2�2K true portfolio risk. (2)

Due to asymmetric information the interpretation of �2 will be di¤erent forthe agent and the principal.The agent is fully informed and knows that �2K is the true idiosyncratic risk

that cannot be diversi�ed, and that there is no systematic risk element whichmight have come from common general economic shocks. Hence the agent alsoknows that the level of the observable portfolio risk �2 depends only on thechoice of portfolio share b of the risky asset.The principal however has no such information about assets, except the ob-

servable portfolio outcome. Under regular conditions the principal would assumethat the observed �2 includes idiosyncratic and systematic elements from gen-eral common shocks which are not connected to the individual portfolio choice.Hence, the principal needs to identify the idiosyncratic risk to asses the speci�cperformance of the managed portfolio. To illustrate the idea this is an example.The principal observes returns of the �rm�s portfolio and calculates the meanas estimator for the expected return as well as the squared standard deviation

7

as estimator for the variance and hence the risk. However, if the principal nowwants to asses the speci�c performance of the protfolio they presume that e.g.a recent global downturn of the economy is an important element of the ob-served deviation of the mean, such that the portfolio might have well performedexcept for the global shock. Therefore, in order to asses the performance oftheir speci�c portfotio they would need information how systematic shocks af-fected the observed outcome. However, by de�nition of asymmetric informationthe principal�s infomration are imperfect, while the agent has full information.This information asymmetry leads to an opportunity for the agent to conductunobservable actions.

The principal is risk averse with an Arrow-Pratt coe¢ cient of absoluterisk aversion �U > 0. To asses the portfolio performance the principal is inter-ested (i) in the returns, and (ii) the identi�cation of idiosyncratic risk perfor-mance of the managed portfolio, because only portfolio speci�c risk is subjectto active choices; systematic risk components generated by common shocks areout of the control of portfolio managers, and hence is independent of the speci�cportfolio structure which is subject to evaluation.Further, the principal can observe only portfolio returns. It neither does

know about the true risk of single assets, nor does it even know what set ofrisky assets exists in the economy, or to what extend common shocks trulya¤ect total performance of the �nancial system. The principal delegates optimalportfolio choice to an agent. It recruits the agent from a competitive agentmarket without search costs. In order to appoint a manager it negotiates theterms of the performance contract with each applicant. It o¤ers a contract witha �xed payment wf and a linear reward � on perceived performance above closecompetitors�portfolios performance. During these negotiations both parties aretrustable in terms of their observable actions. The principal receives a trustableperformance promise by the agent. That is, the principal will later observethe promised expected return and realizes a perceived risk of the portfolio aspromised in the contract.However, for the principal the simple observable portfolio variance is not a

perfect indicator of true idiosyncratic portfolio risk as it might include system-atic risk. The variance is subject to interpretation within a reported context.However, the reported context and the selection of information for this inter-pretation is part of the agent�s job. As the principal can only get hold of a verylimited set of information, the largest and most detailed amount of informationis exclusively owned by the agent. When the principal evaluates the observ-able volatility it will have to use the information and the interpretation of theagent to identify the idiosyncratic portfolio speci�c risk. Hence, the principal�sperception of idiosyncratic risk will depend heavily on the agent�s advice.With reference to the contract and the agent�s credibility concerning risk,

the agent is credible in that the principal�s perception of portfolio risk will notdeviate from the promises made in the contract. With respect to observablereturns and the perception of risk, the agent appears to have met their con-

8

tractual obligations. As a result the principal will hire an agent based on theirpromises which, according to their best information, will also be ful�lled.

The agent also is risk-averse. The Arrow-Pratt coe¢ cient of absoluterisk aversion for the agent is �u > 0. Comparing the agent�s and principal�srisk preferences we suppose that the principal is at least as risk averse as theagent, �U � �u. The agent has perfect information. Upon hiring, they will givea credible promise that they will act according to their contractual obligations.That is, the agent is credible in that the principal�s perception of portfolio per-formance will not deviate from the promises made in the contract. However, theagent simultaneously manipulates information that is important for risk percep-tion and risk evaluation. The agent takes advantage of asymmetric information.That is, they try to hide true idiosyncratic risk and manipulate portfolio speci�cperception. The agent conceals the true risk structure with concealing e¤ort q;marginal concealing costs c, and concealing e¤ect 1=q such that only 1=q ofobservable volatility �K becomes perceived as idiosyncratic risk, even if �K infact indicated the true portfolio speci�c risk. Hence parts of the true portfoliospeci�c risk �K are wrongly related to systematic risk components which trulydo not exist.

Principal-agent interaction is determined by asymmetric informationwith respect to a deviation of the true portfolio speci�c risk and the principal�sperception of this risk. Due to such an information asymmetry and moral hazardthere may be hidden actions, so a destructive agent will use this asymmetry totheir advantage. The idea of modeling economic activities not meant to serveproductively for a market or costumer is not new. The notion of an economicperson that deliberately acts destructively to gain pro�ts was notably broughtup by Baumol (1990) in the context of entrepreneurship. In a di¤erent contextand with a slightly di¤erent background idea, rent seeking addresses a similarissue.6 However, transferring this concept to the principal-agent problem isan obvious step. Pro�ting from hidden actions even if they deliberately harmothers is a behavior that we cannot categorically rule out in business. In ourmodel the agent can allocate e¤ort. While the principal believes that the agent�se¤ort will improve portfolio performance, the agent makes an e¤ort to concealthe true risk and manipulate observable risk indicators.7

2.2 Contract Design

The agent�s decision problem: The agent has a utility function of theform u(y) = �e�f(y) with income y. If the agent is hired by the principal they

6Bhagwati (1982) and later Murphy et al. (1991) and (1993) developed the pro�t and rentseeking concept as non-productive economic activity generating gains for the rent seeker.

7We could of course include positive e¤orts as well and optimally allocate e¤ort betweenconstructive and destructive e¤orts. However, to focus on the point and keep things as simpleas possible, we assume with respect to positive e¤orts that costs of e¤ort towards positivechanges are too high that theses e¤orts are conducted.

9

obtain a �xed salary wf and a symmetric linear performance related incomewp, so that the total income is y = wf + wp. Since portfolio performance is arandom variable the performance related salary wp is also random and has tobe evaluated as such. To determine the "performance"-related reward, we needa benchmark to compare with. Due to the assumption of information asym-metry the principal is a rather uninformed person who does not have detailedinformation or assessment tools about current �nancial conditions and �nancialmarkets. This fact in particular is the reason why the principal delegates wealthmanagement to the agent. Therefore, the most simple way to evaluate the rel-ative performance of the own portfolio is to look at competitors in the market.Close competitors have similar business models and returns of competing �rmsare public knowledge. Hence the principal asseses the own portfolio performancerelatively to the performance of the closest competitors of the �nance �rm. Theprincipal rewards the agent if the perception of the own managed portfolio Rperforms better than the portfolios of close competitors whose returns RCC arealso normally distributed.Hence, if the principal gives a proportional reward � for ostensibly good per-

formance [wp = � [R�RCC ]] we can describe a symmetric linear performancecontract by

wf + � [R�RCC ] :

The agent�s utility function can now be written as

u(y) = �e��u(wf+�[R�RCC ]):

Since we assume that the return of the risky asset follows a normal distrib-ution the agent�s expected utility would be8

Eu = �e��u

�wf+�ER�

�u�2b2�2K2 �E[�RCC ]+

�u�2�2CC2

�:

Further, under perfect information or perfect loyalty the agent could nowchoose a portfolio share that maximizes their expected utility and simultane-ously also the principal�s. However, while the principal believes that the agentis loyal, the agent is aware of information asymmetries and acts destructively.That is, the principal is able to observe portfolio returns, yet they cannot ob-serve the true idiosyncratic risk. They cannot use the observable variance astrue risk indicator. They presume that the observable variance contains usualidiosyncratic and systematic risk elements. However, for the assessment of theportfolio performance they need to �lter the idiosyncratic component. Hence,with respect to idiosyncratic risk assessment of the portfolio the principal hasto rely on interpretations and indicators that are reported by its agent. Hence,the destructive agent can manipulate and reduce the principal�s perception ofspeci�c portfolio related risk. The agent can increase the portfolio share b of

8Here, ERM is the expected return of the market portfolio and �2M is the variance of themarket portfolio indicating market risk. Both, the expected market return and the marketrisk, is public knowledge.

10

the risky asset and simultaneously reduce the principal�s perception of idiosyn-cratic risk by concealing e¤orts. In the model, the agent�s risk-concealing e¤ortq � 1 a¤ect the principal�s portfolio speci�c risk perception by 1=q. However,concealing risk comes at a price. Marginal costs increase with the share of therisky asset b2, and depend on the agent�s individual concealing cost factor c. Ifan agent can easily manipulate information with very little cost to themselves,c is small. By de�nition, the destructive agent su¤ers no additional costs whenbeing disloyal or even actively harming the principal. They are purely actingin their own interest with full awareness of information asymmetries. The de-structive agent�s total concealing costs are then given by cb2q. Implementingthe destructive agent-related elements in the model, the perceived portfolio spe-ci�c risk of the principal is reduced by 1=q of the observable volatility and theconcealing e¤ort reduces the agent�s income by cb2q. The agent�s expected util-ity can now be maximized by two choices, the choice of the optimal concealinge¤ort q and the choice of the optimal portfolio share b. Maximization is subjectto a contract o¤ered by the principal with a �xed payment wf and a symmetricperformance factor � as well as the rationality constraint.

maxb;qEu(y) = max

b;q� e

��u�wf�cqb2+�ER�

�u�2b2�2K2q ��ERCC+

�u�2�2CC2

�(3)

s.t. u ( �w) � Eu(y): (r.c.) (4)

The rationality constraint (4) states that the utility of the outside optionu ( �w) is not higher than the expected utility of the contract Eu(y).

Principal�s Decision Problem: The principal has a utility function ofthe form U(Y ) = �e�F (Y ) with income Y: Net income consists of asset returnsfrom the �rms portfolio R minus a �xed salary wf and a performance-relatedsalary wp paid to the agent who manages the portfolio. The principal comparesthe perceived performance of the managed portfolio R with given portfoliosof close competitors RCC and rewards the agent in a linear and symmetricmanner for the excess performance, � (R�RCC).9 Hence, its income is Y =R� wf � � (R�RCC) : We can specify the principal�s utility function as

U = �e��U (R�wf��(R�RCC)):

The expected utility for a normally distributed return of the own and com-petitors�portfolios is

EU = �e��U

�(1��)ER+�ERCC�wf�

�U (1��)2b2�2K2 � �U�2�2CC

2

�9Again, as mentioned before, market performance is public knowledge.

11

It is worth noting again that due to information asymmetries the principalcannot observe the true idiosyncratic risk of the portfolio. It trusts its agentand relies on the agent�s information. However, the reported portfolio speci�crisk is manipulated with concealing e¤ort q and concealing e¤ect 1=q. Theprincipal maximizes expected utility considering the rationality constraint (4)and incentive constraint (6) by choosing the contract components wf and �:

max�;wf

EU(�;wf ) = max�;wf

� e��U

�(1��)ER+�ERCC�wf�

�U (1��)2b2�2K2q � �U�2�2CC

2

�(5)

s.t (b; q) = (b�; q�) 2 argmaxb;qEu(b; q), (i.c.) (6)

and u(_w) � E(u(y)) (r.c.)

Drawing up a contract: Having identi�ed the decision problems we cannow determine all elements of a contract. That is, when a principal announcesa vacancy an agent will apply and negotiate the terms and conditions of theircontract with the principal. The result of these negotiations is a potentialagreement on a contract as de�ned in Proposition 1.

Proposition 1 (contract): With asymmetric information and hidden actionthere exists a potential contract under which (i) the agent chooses an optimalportfolio share of the risky asset b� and an optimal risk concealing e¤ort q� giventhe principal�s o¤er of a �xed salary wf and a performance share �

b� = b� (�u; c; ERK ; ERB ; �K) (7)

q� = q� (�u; c; �; �K) : (8)

(ii) the principal chooses an optimal �xed salary and an optimal performanceshare depending on perceived performance.

�� = �� (�U ; �u; c; ERK ; ERB ; �K ; �CC) ; withd�

d�U< 0 (9)

w�f = w�f (�U ; �u; c; ERK ; ERB ; �K ; �CC) : (10)

For a proof of Proposition 1 see Appendix 1.

2.3 Adverse selection of agents

So far we have determined a contract between a principal and a potential agentgiven the characteristics of both parties. In this model the principal is com-pletely characterized by (i) its risk preferences (risk averse), (ii) its objective toperform better than competitors and (iii) its preference for a symmetric perfor-mance contract. Since we assume that all these characteristics are identical for

12

all principals the modeled principal is representative of the economy. Agents dif-fer in their individual risk preferences �uj 2 R and concealing costs cj 2 R. If anagent is able to easily manipulate information and conceal true risks, cj will besmall. An agent who is not willing or unable to manipulate information will havehigh concealing costs cj . In this model we assume that concealing costs and thedegreee of risk aversion is distributed among heterogeneous agents such that wecan order all n agents according to the degree of risk aversion [�u1 < �uj < �un]and/or their concealing costs c1 < cj < cn.When the principal o¤ers a vacancy a number of prospective agents will ap-

ply and the principal will negotiate with each of the candidates. A potentialcontract is negotiated with each applicant and the principal can choose fromamong the applicants according to their promises and to the apparent utilitiesthese promises would generate. As described above, the agent ful�lls the termsof their contract according to the principal�s perceived (but manipulated) infor-mation. Then the principal chooses the agent whose contract seems to generatethe highest (indirect) utility under the contract. This leads to the followingproposition.

Proposition 2 (principal�s utility from agent�s attributes): When the principalrecruits a potential agent (i) ostensible (indirect) utilities from perceived perfor-mance increase with agent�s having decreasing risk aversion �u; or decreasingconcealing costs c.

@EU� (�u)

@�u< 0 for �u 2 (0,NEU�);

@EU� (c)

@c< 0 for su¢ ciently small �U

(ii) True (indirect) utilities from true �rm performance decrease with decreasingrisk aversion �u; or decreasing concealing costs c

@EU�true(�u)

@�u> 0 , for su¢ ciently large �CC

@EU�true(c)

@c> 0 , for su¢ ciently small c.

For a proof of Proposition 2 see Appendix 2.

This proposition makes clear that the activities of a destructive agent willdeceive the principal and cause harm and hence are ine¢ cient. The more un-scrupulous an agent is when concealing risk, the better they seem to perform andthe greater the damage to the principal.What is more, the more risk loving anagent is, the more successful they seem to be in the perception of the principal.Technically speaking, the agent deliberately produces a negative external e¤ectfor the principal which they can hide. This e¤ect is not adequately rewarded inthe contract and is hence a (currently unobservable) negative externality that

13

a¤ects the principal. The contract fails to generate an e¢ cient exchange ofservice and reward. The agent is rewarded for a service they do not deliver.However, the misconception about the agent�s real performance has another,

even worse dynamic e¤ect. Once a principal recruits an agent and negotiatesthe contracts it will be misguided. Because of asymmetric information, it willprefer less honest and more unscrupulous or more risk loving agents. This leadsto the next proposition.

Proposition 3 (principal�s adverse selection of agent): (i) For a set of compet-ing agents that di¤er only in terms of the degree of risk aversion �u or concealingcosts c, the principal chooses the agent with the� lowest risk aversion �minu and/or the lowest concealing costs cmin

��u = �minu since@EU�(�u)

@�u< 0

c� = cmin since@EU�(c)

@c< 0

� largest share of risky assets and hence the most risky portfolio �2

b� = bmax = b(c�; ��u) since@b

@c< 0 and

@b

@�u< 0 ;

�2max = (bmax)2�2K

(ii) This choice implies the principal�s lowest expected "true (indirect) utility"

EU�mintrue = EU�true(��u; c

�) since@EU�true(�u)

@�u> 0;

@EU�true(c)

@c> 0

For a proof of Proposition 3 see Appendix 3.

This proposition describes an adverse selection in the market for agents. Thee¤ects of this adverse selection of agents refers back to the risk behavior of the�rm.The lower an agent�s risk aversion �u; the less painful risk taking for the

agent. Further, the lower concealing costs cj of an agent, the less costly thisagent�s concealing e¤orts. Agents with low concealing costs do not need to spendmuch when conducting concealing activities, and hence concealing risk generateseasy pro�ts. Furthermore, even if agents are only a "little bit" disloyal, suchthat their concealing costs are not prohibitive but still quite high, the principalwould nevertheless choose the agent with the lowest risk aversion they can �nd.That is, even if in a rather honest society adverse selection of unscrupulus agents

14

soon terminates, the selection mechanism would still pick the agent with lowestrisk aversion among the less scrupulus agents. As a result, selecting the wrongagent leads to excessive risk in the portfolio of each individual �nance �rm. Risk-taking is ine¢ cient because a �rm�s principal would not choose this portfoliostructure if it knew about the true risk of the portfolio. Moreover, since adverseselection is driven by a race to �nd the agent with the lowest risk aversion orconcealing costs, the principal will eventually choose the agent who constructsthe most risky portfolio possible. Hence each �nance �rm will choose the highestpossible risk.

3 Financial Market Failure & Systemic Risk

3.1 Market Portfolio and Aggregate Risk:

Having discussed the micro-perspective in a given �nance �rm and individual�rm selection of agents, we now need to look at the aggregate e¤ects. This seemsstraightforward as we have already established that performance contracts andthe behavior associated with them are dominant in or even representative ofthe considered market. Corollary 4 states the implications of the above microsetting for the aggregate market if �rms o¤er performance contracts to managerswho manage their �rms�portfolio choice.

Corollary 4 (aggregate market portfolio and risk): If all �rms o¤er perfor-mance contracts with rewards for out-performing close conmpetitors� portfo-lios, all principals choose the agent with attributes cmin and �minu , and hencethe market portfolio becomes the most risky portfolio possible in the economy[�2 = (bmax)2�2K for all �rms] .

For a proof see Appendix 4.

This corollary simply summarizes individual �ndings for the aggregate mar-ket. Since competitors�portfolios are the benchmark and principals in each �rmwill try to beat competitors and sets incentives each �rm is driving the markettowards a higher risk. As a result, setting the wrong incentives in performancecontracts and picking the wrong agents will not just lead to excessive risk inone �rm; rather, the entire sector will take risks at an ine¢ cient and excessivelevel.

3.2 Destructive Agents and Systemic Risk:

Having described the main mechanism that leads to individual and aggregaterisk-taking the link to the term systemic risk can be drawn. Systemic riskis not a clearly and uniformly de�ned term, and it is used in various ways.10

Often it is related to contagion and hence describes an externality occurring

10For a broad discussion see Hellwig (2009).

15

often exogenous under certain macro conditions. However, in this model wewould like to use a de�nition of systemic risk that directly relates to the modelmechanics introduced above and gives a consistent and endogenous explanationof when it occurs and when it does not.

De�nition 5 (systemic risk): Systemic risk is an ine¢ cient and excessive riskas stable outcome of an aggregate market, directly resulting from endogenouschoices of market participants.

Clearly, this rather narrow de�nition cannot accommodate the many phe-nomena the term sometimes relates to. However, it attempts to relate the termto a structure that endogenously promotes excessive risk due to ill-designed�rms or market characteristics and mechanisms or wrong incentives that causebad behavior throughout the whole system. In other words, systemic risk is anexcessive risk that endogenously evolves out of badly designed conditions anda¤ects the entire system. Hence we can state the following theorem:

Theorem 6 (destructive agents and systemic risk): If destructive agents existin this delegated portfolio model, performance contracts designed to beat closecompetitors lead to systemic risk.

For a proof see Appendix 6.

The interpretation of this theorem is straightforward. (i) If we extend theactivities of agents from just doing good or better to explicitly and deliberatelyharming others, conditions such as asymmetric information and moral hazardbecome an even more serious threat. Ine¢ ciency and market failure are likelyto occur. In the discussed case they can even destabilize an entire sector or eventhe economy at large. Technically, negative externalities are not just side e¤ectsthat are ignored by the originator; they are generated on purpose as they arethe only pro�t making activity.(ii) As for the argument of limited liability being a major reason for excessive

risk-taking, we reveal an aspect that is not yet in the focus of academic research.Limited liability within a �rm due to publicly unobservable information is amajor source of wrong incentives for agents. As a publicly unobservable riskburden can be allocated to principals with only a limited e¤ect for agents, theycan exploit this limited liability for their own purpose.(iii) Further, the ownership of managed capital does not matter. As soon

as the portfolio is delegated to an agent, the agent may behave destructivelyand allocate unobservable risk to the capital owner while obtaining rewardsfor observable returns. This is independent of the leverage. In our model theprincipal owns one hundred percent of the invested capital. Hence, within thisscenario the share of equity does not matter and an increase in this share, asrecently suggested in the political debate, will not take risk out of the system.(iv) High risk taken by all �nance �rms implies instability if risky events

are realized. Badly designed �rms and misunderstood incentive schemes within

16

�rms can become a permanent source of instability not only for the �rm inquestion. If this kind of bad �rm design were to become the dominant model ina given sector, that entire sector would become a source of instability.

4 Summary and Conclusions

We investigate the popular claim made after the 2008 �nancial crash that a mal-functioning, poorly designed incentive scheme in �nancial �rms that encouragedgreed and involved excessive salaries was responsible for excessive risk takingand eventually led to the crash. In this context three questions arise. (i) Canasymmetric information, moral hazard, and hidden action in �nance �rms ex-plain excessive risk-taking? That is, does the principal-agent problem matter inthis context? More precisely, (ii) how do contractual incentives cause excessiverisk-taking at individual �rm level? And �nally, (iii) how do malfunctioningincentives within �nance �rms cause systemic risk at sector level?To answer these questions we use a delegated portfolio choice approach with

performance contracts. In addition to the existing discussion, but in analogy toBaumol�s destructive entrepreneur,11 we introduce the destructive agent. Thedestructive agent does not simply adjust to given asymmetric information; theydeliberately generate wrong information for their own advantage and at theexpense of the principal. Thanks to this actively manipulated and asymmetricinformation about the true idiosyncratic portfolio risk, the agent appears toperform well and hence is paid a high performance-related salary.As these activities are unobservable for the principal, an ability to deceive

and manipulate information becomes crucial for the ostensible performance ofan agent. The lower the agent�s risk aversion, or the better they are at con-cealing portfolio speci�c risk information, the better they seem to perform andthe better �ostensibly �this is for the principal. An adverse selection processfor agents leads to the employment of the greediest applicant. Hence, in thepresence of performance contracts and destructive agents, a �rm will take ex-cessive risk �more than the principal would like to take. However, the failure ofan individual �rm translates into excessive risk for the entire sector if all �rmsapply performance contracts and if they reward outperforming of close competi-tors�portfolios. Having de�ned systemic risk as an ine¢ cient and excessive riskthat is endogenously chosen by market participants and a stable phenomenonin the aggregate market, we illustrate how a malfunctioning and badly designedincentive system can endogenously destabilize the �nancial market system andimply systemic risk.Hence, if we allow for the existence of deliberately harming agents in our

model - while maximizing their private utility - we obtain under the discussedconditions a general market failure with respect to risk taking.

11See Baumol (1990).

17

References

[1] Admati, A. R. & P�eiderer, P., 1997. Does it all add up? Benchmarksand the Compensation of Active Portfolio Managers. Journal of Business,70(3): 323�350.

[2] Baumol, W.J., 1990. Entrepreneurship: Productive, Unproductive, andDestructive. Journal of Political Economy, 98(5): 893-921.

[3] Bhagwati, J.N., 1982. Directly Unproductive, Pro�t-Seeking (DUP) Activ-ities. Journal of Political Economy, 90(5): 988-1002.

[4] Bhattacharya, S. & P�eiderer, P., 1985. Delegated Portfolio Management.Journal of Economic Theory, 36(1): 1-25.

[5] Biais, B. & Casamatta, C., 1999. Optimal Leverage and Aggregate Invest-ment. Journal of Finance, 54(4): 1291-1323.

[6] Carpenter, J. N., 2000. Does Option Compensation Increase ManagerialRisk Appetite? Journal of Finance, 55(5): 2311�2331.

[7] Chen, H. L. & Pennacchi, G., 2002. Does Prior Performance A¤ect a MutualFund�s Choice of Risk? Theory and Further Empirical Evidence. WorkingPaper, University of Illinois.

[8] Das, S. R. & Sundaram, R. K., 2002. Fee Speech: Signaling, Risk Sharing,and the Impact of Fee Structures on Investor Welfare. Review of EconomicStudies, 15(5): 1465�1497.

[9] Dash, E. & Creswell, J., 2008. Citigroup Pays for a Rushat Risk. New York Times, November 23. available at:http://www.nytimes.com/2008/11/23/business/worldbusiness/23iht-23citi.18059343.html?pagewanted=all [Accessed September 21, 2013].

[10] Desai, M., 2012. The Incentive Bubble. Harvard Business Review, 90(3):124-132.

[11] Gollier, C., Koehl, P. F. & Rochet, J. C., 1997. Risk-Taking Behavior withLimited Liability and Risk Aversion. Journal of Risk and Insurance, 64(2):347�370.

[12] Grinblatt, M. & Titman, S., 1989. Portfolio Performance Evaluation: OldIssues and New Insights. Review of Financial Studies, 2(3): 393-421.

[13] Hellwig, M., 1994. Liquidity Provision, Banking, and the Allocation ofInterest Rate Risk. European Economic Review, 38(7): 1363-1389.

[14] Hellwig, M., 2009. Systemic Risk in the Financial Sector: An Analysis ofthe Subprime-Mortgage Financial Crisis, De Economist, 157, (2): 129-207.

18

[15] Holmstrom, B. & Milgrom, P., 1987. Aggregation and Linearity in theProvision of Intertemporal Incentives. Econometrica, 55(2): 303�328.

[16] Li, C.W. & Tiwari, A., 2009. Incentive Contracts in Delegated PortfolioManagement. The Review of Financial Studies, 22(9): 4681-4713.

[17] Murphy, K.M., Shleifer, A. & Vishny, R. W., 1991. The Allocation of Tal-ent: Implications for Growth. Quarterly Journal of Economics, 106(2): 503-530.

[18] Murphy, K.M., Shleifer, A. & Vishny, R. W., 1993. Why Is Rent-Seekingso Costly to Growth? American Economic Review, 83(2): 409-414.

[19] Palomino, F. & Prat, A., 2003. Risk Taking and Optimal Contracts forMoney Managers. RAND Journal of Economics, 34(1): 113-137.

[20] Ross, S., 2004. Compensation, Incentives, and the Duality of Risk Aversionand Riskiness. Journal of Finance, 59(1): 207�225.

[21] Sheng, J., Wang, X. & Yang, J., 2012. Incentive Contracts in DelegatedPortfolio Management under VaR Constraint. Economic Modelling, 29(1):1679-1685.

[22] Starks, L. T., 1987. Performance Incentive Fees: An Agency TheoreticApproach. Journal of Financial and Quantitative Analysis, 22(1): 17�32.

[23] Stoughton, N.M., 1993. Moral Hazard and the Portfolio Management Prob-lem. The Journal of Finance, 48(5): 2009-2028.

[24] Stracca, L., 2006. Delegated Portfolio Management: A Survey of the The-oretical Literature. Journal of Economic Surveys, 20(5): 823-848.

19

5 Appendix

5.1 Appendix 1: Proof of Proposition 1: Drawing up acontract

5.1.1 (i) Proof of the agent�s optimal choice

The utility function of an agent is given by:

u(y) = �e��u(wf+�(R�RCC))

De�nition of assets gives:

R = (1� b)RB + bRKE [R] = : ER = E [(1� b)RB + bRK ] = (1� b)ERB + bERK

V ar [R] = �2R = V ar [(1� b)RB + bRK ] = (1� b)2V ar [RB ] + b2V ar [RK ]= b2�2K

If R � N (ERK ; �K) and RCC � N (ERCC ; �CC) are normally distributed weobtain for the expected utility

E [u(y)] = : Eu = E��e��uwf

�� E�e���uR+��uRCC

�= �e

��u�wf+�ER�

�u�2b2�2K2 �E[�RCC ]+

�u�2�2CC2

�

When the agent manipulates information we have to add the factor of 1q as

manipulation of variance perception and the manipulation costs cqb2. As aresult the agent maximizes expected utility:

Eu = �e��u

�wf�cqb2+�ER�

�u�2b2�2K2q ��ERCC+

�u�2�2CC2

�

Continuity of the exponential function leads to the equivalent maximizationproblem:

maxb;qF (b; q) : = max

b;q

�wf � cqb2 + �ER�

�u�2b2�2K2q

� �ERCC +�u�

2�2CC2

�s.t. u(

_w) � E(u(y))

For this problem the FOCs are:

0 =@F (b; q)

@b= 2cqb+ �ERB � �ERK + �u

�2b�2Kq

0 =@F (b; q)

@q= �cb2 + �u

�2b2�2K2q2

20

As a result, it follows:

) c = �u�2�2K2q2

= c�

) q2 = �u�2�2K2c

;) q =

p�u��Kp2c

= q�

and

0 = 2cb

p�u��Kp2c

+ �ERB � �ERK + �u�2b�2Kp�u��Kp2c

, b =ERK � ERB2p2c�u�K

=: b�

Now we need to prove that b� and q� are really maxima, therefore we takea look at the second derivatives and the Hesse-Matrix.

@2F (b; q)

@b2=

��2cq � �u�

2�2Kq

�= �2cq � �u�

2�2Kq

@2F (b; q)

@q2=

���u�

2b2�2K4q3

�= ��u�

2b2�2K4q3

@2F (b; q)

@b@q=

@2F (b; q)

@b@q=

��2cb+ �u�

2b�2Kq2

�= �2cb+ �u�

2b�2Kq2

Hence it follows:

@2F (b�; q�)

@b2= �2c

p�u��Kp2c

� �u�2�2Kp

�u��Kp2c

= �2p2c�u��K

@2F (b�; q�)

@q2= �

�u�2

�(ERK�ERB)

2p2c�u�K

�2�2K

4�p

�u��Kp2c

�3 = �p2c(ERK � ERB)216p�u��

3K

@2F (b�; q�)

@b@q=

@2F (b�; q�)

@q@b= �2c (ERK � ERB)

2p2c�u�K

+�u�

2 (ERK�ERB)

2p2c�u�K

�2K�p�u��Kp2c

�2= �

p2c(ERK � ERB)2p�u�K

+p2c(ERK � ERB)2p�u�K

= 0

H(b�; q�) =

�2p2c�u��K 0

0 �p2c(ERK�ERB)

2

16p�u��

3K

!and taken as a conclusion from the de�nition of a de�nite matrix, H(b�; q�)

is negative de�nit.

21

5.1.2 (ii) Proof of the principal�s optimal choice

With the principal�s utility function

U = �e��U (R�wf��(R�RCC))

we analogously obtain the principal�s expected utility

EU = �e�Uwf � e��U

�(1��)ER+�ERCC�

�U (1��)2b2�2K2 � �U�2�2CC

2

�!:

Including the term that manipulates the perception of information in the prin-cipal�s utility 1

q , we obtain the principal�s expected utility, which he wants tomaximize:

EU = �e��U

�(1��)ER+�ERCC�wf�

�U (1��)2b2�2K2q � �U�2�2CC

2

�

Continuity of the exponential function leads to the equivalent maximizationproblem:

max�;wf

EU(�;wf ) = max�;wf

(1� �)ER+ �ERCC � wf �

�U (1� �)2b2�2K

2q� �U�

2�2CC2

!s.t. u(

_w) � E(u(y))

and (b; q) = (b�; q�) 2 argmaxb;qEu(b; q)

Therefore the following Lagrange-function is given by

LU : =

(1� �)ER+ �ERCC � wf �

�U (1� �)2b2�2K

2q� �U�

2�2CC2

!+�U

�u(_w)� E(u(y)�)

�

=

(1� �)ERB + (1� �) b (ERK � ERB) + �ERCC � wf �

�U (1� �)2b2�2K

2q� �U�

2�2CC2

!

+�U

"u(_w) + e

��u�wf�cqb2+�(1�b)ERB+�bERK�

�u�2b2�2K2q �E[�RCC ]+

�u�2�2CC2

�#

=

(1� �)ERB +

��5�+ 6� 1

a

�(ERK � ERB)2

8p2c�u�K

+ �ERCC � wf ��U�

2�2CC2

!

+�U

"u(_w) + e

��u�wf+

2�(ERK�ERB)2

8p2c�u�K

+�ERB�E[�RCC ]+�u�

2�2CC2

�#

22

The FOCs are@LU@�

= 0 =@LU@wf

= 0 =@LU@�U

Therefore it follows:

0 =@LU@wf

= �1 + �U (��u) (�Eu�))1

�uEu� = �U

0 =@LU@�U

= u(_w)� E(u(y))� ) u(

_w) = Eu�

0 =@LU@�

=

�ERB +

��5 + 1

a2

�(ERK � ERB)2

8p2c�u�K

+ ERCC � �U��2CC

!

+�U

2 (ERK � ERB)2

8p2c�u�K

+ ERB � E [RCC ] + �u��2CC

!� (��u) (�Eu�)

0 = �3a2 (ERK � ERB)2

8p2c�u�K

+(ERK � ERB)2

8p2c�u�K

� (�U � �u)�3�2CC

Using the derivative @LU@�U

, we obtain an implicit equation that has to besolved for wf :

u( �w) = Eu�

, ln(�u( �w)) = ��uwf �2�p�u(ERK � ERB)2

8p2c�K

+ ��uERB + ��uERCC +�2u�

2�2CC2

, 0 = ��uwf �2�p�u(ERK � ERB)2

8p2c�K

+ ��uERB + ��uERCC +�2u�

2�2CC2

� ln(�u( �w))

Because no explicit solution for � can be derived we need to show thatimplicit solutions exist.

F :=

�F1F2

�:=

� �3�2 (ERK�ERB)2

8p2c�u�K

+ (ERK�ERB)2

8p2c�u�K

� (�U � �u)�3�2CC

��uwf �2�p�u(ERK�ERB)2

8p2c�K

+ ��uERB + ��uERCC +�2u�

2�2CC2 � ln(�u( �w))

�

Therefore, it follows F = 0 with notice to the FOCs. The matrix

@F

@(�;wf )=

0@ �6� (ERK�ERB)2

8p2c�u�K

� 3(�U � �u)�2�2CC 0

�p�u(ERK�ERB)

2

4p2c�K

+ �uERB + �uERCC + �2u��

2CC ��u

1A23

is invertible, because

det(@F

@(�;wf )) = 6�

p�u (ERK � ERB)

2

8p2c�K

+ 3(�U � �u)�u�2�2CC 6= 0

thus the implicit function theorem can be applied and we obtain:

�� = �� (�U ; �u; c; ERK ; ERB ; �K ; �CC) ;

with@�

@�U= � �2�2K

(6 (ERK�ERB)2

8p2c�u�K

+ 3(�U � �u)��2CC)< 0; for �U � �u

with@�

@c=

��3�2 � 1

� (ERK�ERB)2

16p2c�uc�K

��6� (ERK�ERB)

2

8p2c�u�K

+ 3(�U � �u)�2�2CC�

@�

@c< 0 if � <

1p3and

@�

@c> 0 if � >

1p3

w�f = w�f (�U ; �u; c; ERK ; ERB ; �K ; �CC) :

Conclusion:Then Proposition 1 leads to the optimal choices:

b� = b� (�u; c; ERK ; ERB ; �K) =ERK � ERB2p2c�u�K

q� = q� (�u; c; �; �K) =

p�u��Kp2c

�� = �� (�U ; �u; c; ERK ; ERB ; �K ; �CC)

w�f = w�f (�U ; �u; c; ERK ; ERB ; �K ; �CC)

where �� becomes the smaller, the larger the risk aversion of the principal�U �

5.2 Appendix 2: Proof of Proposition 2: principal�s utilityand agent�s attributes

5.2.1 Ostensible (indirect) utility reaction:

a) Now it is to show for the ostensible (indirect) utilities, @EU�(�u)

@�u< 0 , where

EU� is given as:

EU� = �e��U

�(1���)ER+��ERCC�w�f�

�U (1���)2(b�)2�2K2q� � �U (��)2�2CC

2

�

24

To obtain this result we use the envelope theorem and determine the deriv-ative of the Lagrange function at �� and w�f .Proof.

@LU@�u

=

��5�+ 6� 1

�

�(ERK � ERB)2

8p2c�K

��12

1

�up�u

�+�U (�Eu�)

��wf �

2�(ERK � ERB)2

8p2c�K

��12

1p�u

�� �ERB + �ERCC �

2�u�2�2CC2

�=

�5�� 6 + 1

�

�(ERK � ERB)216p2c�u�K

1

�u

+1

�u

�wf �

2�(ERK � ERB)216p2c�u�K

+ �ERB � �ERCC + �u�2�2CC�

=1

�u

��3�� 6 + 1

�

�(ERK � ERB)216p2c�u�K

+ [wf + �ERB � �ERCC ]�+ �2�2CC

gives us the derivative of the utility function with concealing costs:

@EU�

@�u=1

�u

��3�� 6 + 1

�

�(ERK � ERB)216p2c�u�K

+ [wf + �ERB � �ERCC ]�+�2�2CC

At �rst, we want to analyse the algebraic sign of the �rst obtained derivative,and in order to keep the arguments clear, we de�ne some abbreviations

x : =

�3�� 6 + 1

�

�(ERK � ERB)2

16p2c�K

y : = [wf + �ERB � �ERCC ]z : = �2�2CC

We note that these abbreviations take on the following algebraic signs: x < 0,y > 0 and z > 0. We can now shorten @EU�

@�uto a function f , de�ned as follows:

f(�u) :=x

�up�u+y

�u+ z

this function is now only dependent on the variable of interest, �u, and possessesthe same characteristics as @EU�

@�u.

A closer look at the limits

lim�u �!1

f(�u) = lim�u �!1

x

�up�u+y

�u+ z = z > 0

lim�u �!0

f(�u) = lim�u �!0

x

�up�u+y

�u+ z = �1 < 0

shows that f owns at least one root in (0;1). With further arguments ofpolynomial algebra, we can deduce that f owns exactly one root Nf in (0;1),and inserting the terms for x, y and z yields a root NEU� of @EU

�

@�uin (0;1).

25

Hence, there exists an interval, where the �rst derivative of the EU� functionis negative.b) Now it is to show for the ostensible (indirect) utilities, @EU

�(c)@c < 0 .

To obtain this result we use the envelope theorem and determine the deriv-ative of the Lagrange function at �� and w�f .

@LU@c

= �12

��5�� + 6� 1

��

�(ERK � ERB)2

8p2�u�K

c�32

+�U

24e��u�w�f+ 2��(ERK�ERB)2

8p2�u�K

c�12+�ERB�E[�RCC ]+

�u(��)2�2CC

2

�| {z }

�Eu�

35 �u2

2�� (ERK � ERB)2

8p2�u�K

c�32

= �12

��5�� + 6� 1

��

�(ERK � ERB)2

8p2�u�K

c�32 + �U (�Eu�)

�u2

2�� (ERK � ERB)2

8p2�u�K

c�32

@EU� (c)

@c= �1

2

(ERK � ERB)2

8p2�u�K

c�32

��3�� + 6� 1

��

�< 0 for su¢ ciently large �� � 1�

�3�� + 6� 1

��

�> 0 if �� > 1�

r2

3� 0; 1835

Hence we obtain for the ostensible (indirect) utilities EU�(c) with@EU�(c)

@c < 0 for su¢ ciently large �� � 1.

5.2.2 True expected (indirect) utility reaction:

a) Even if ostensible (indirect) utilities decrease with c, true utility increase withincreasing �u , thus we need to show that

@EU�true(�u)@�u

> 0, again by using theenvelope theorem for

EU�true = �e��U

�(1���)ER+��ERCC�w�f�

�U (1���)2(b�)2�2K2 � �U (��)2�2CC

2

�

Whereas we can obtain the derivative of the utility function without con-cealing costs from the following Lagrance function:

@LUtrue@�u

= �U (�Eu�)��wf + cqb2 � �(1� b)ERB � �bERK +

2�u�2b2�2K2q

+ �ERCC �2�u�

2�2CC2

�=

1

�u

�wf � cqb2 + �(1� b)ERB + �bERK �

�u�2b2�2Kq

� �ERCC + �u�2�2CC�

=1

�u

�wf � cqb2 + �(1� b)ERB + �bERK � �ERCC

�+ �2

��2CC �

b2�2Kq

�

26

and by using the Envelope Theorem once more, we get

@EU�true@�u

=1

�u

�wf � cqb2 + �(1� b)ERB + �bERK � �ERCC

�+�2

��2CC �

b2�2Kq

�Proof. The algebraic sign of the derivative @EU�

true

@�uneeds a slightly di¤erent

approach, but �rst, we also take a look at the limits at the boundaries for �u:

lim�u�!1

@EU�true@�u

= �2��2CC �

b2�2Kq

�lim

�u�!0

@EU�true@�u

= 1

Since we can manipulate the size of �2CC , the �rst limit can be taken aspositive, and therefore the analysis of the algebraic sign once again centersaround �nding roots for @EU�

true

@�u, it holds:

@EU�true@�u

=1

�u

�wf � cqb2 + �(1� b)ERB + �bERK � �ERCC

�+ �2

��2CC �

b2�2Kq

�= 0

, �u = ��wf � cqb2 + �(1� b)ERB + �bERK � �ERCC

��2h�2CC �

b2�2Kq

iWe label this root as NEU�

true, and take a closer look at the algebraic sign:

NEU�true

=��wf � cqb2 + �(1� b)ERB + �bERK � �ERCC

��2h�2CC �

b2�2Kq

i

=

��2�(ERK�ERB)

2

8p2c�u�K

+ 2�ERB +�u�

2�2CC2 � ln(�u( �w))

�u� �(ERK�ERB)

2

8p2c�u�K

+ 2�(ERK�ERB)2

8p2c�u�K

��2��2CC �

(ERK�ERB)2

4�p2c�u�u�K

�

=

��3�(ERK�ERB)

2

8p2c�u�K

+ 2�ERB +�u�

2�2CC2 � ln(�u( �w))

�u

��2��2CC �

(ERK�ERB)2

4�p2c�u�u�K

�

=�h34�b (ERK � ERB) +

2ERB

� +�u�

2CC

2 � ln(�u( �w))�2�u

ih�2CC � 1

4�b(ERK�ERB)

�u

iIn both bracket terms, �2CC can now be manipulated in its size, so that both

terms are positive, and thus NEU�true

< 0 and @EU�true

@�udoes not own a root in

(0;1), and therefore it follows:

@EU�true@�u

> 0 8 �u > 0

27

b) Even if ostensible (indirect) utilities decrease with c, true utility increasewith increasing c , thus we need to show that @EU

�true(c)@c > 0, again by using the

envelope theorem for

LUtrue =

(1� �)ERB + (1� �) b (ERK � ERB) + �ERCC � wf �

�U (1� �)2b2�2K

2� �U�

2�2CC2

!

+�U

"u(_w) + e

��u�wf�cqb2+�(1�b)ERB+�bERK�

�u�2b2�2K2q �E[�RCC ]+

�u�2�2CC2

�#

=

0@ (1� �)ERB + (1��)(ERK�ERB)2

2p2c�u�K

+ �ERCC � wf

��U (1��)2(ERK�ERB)2

16c�u� �U�

2�2CC2

1A+�U

"u(_w) + e

��u�wf��

(ERK�ERB)2

8p2c�u�K

+�ERB+�(ERK�ERB)2

2p2c�u�K

�� (ERK�ERB)2

8p2c�u�K

�E[�RCC ]+�u�

2�2CC2

�#

=

0@ (1� �)ERB + (1��)(ERK�ERB)2

2p2�u�K

c�12 + �ERCC � wf

��U (1��)2(ERK�ERB)2

16�uc�1 � �U�

2�2CC2

1A+�U

"u(_w) + e

��u�wf+�

(ERK�ERB)2

4p2�u�K

c�12+�ERB�E[�RCC ]+

�u�2�2CC2

�#

@LUtrue@c

=

"(1� �) (ERK � ERB)2

2p2�u�K

(�12c�

32 )� �U

(1� �)2 (ERK � ERB)2

16�u

��c�2

�#

+�U

"(�Eu�) (��u)

� (ERK � ERB)2

4p2�u�K

��12c�

32

�!#

=

"� (1� �) (ERK � ERB)

2

4p2�u�K

(c�32 ) + �U

(1� �)2 (ERK � ERB)2

16�u

�c�2�#

+�U

"(�Eu�) (��u)

�� (ERK � ERB)

2

8p2�u�K

�c�

32

�!#

=

"� (1� �) (ERK � ERB)

2

4p2�u�K

c�32 + �U

(1� �)2 (ERK � ERB)2

16�uc�2

#

+

� (ERK � ERB)2

8p2�u�K

c�32

!

=

"� (ERK � ERB)

2

p2�u�K

c�32

�1

4� �8

�+ �U

(1� �)2 (ERK � ERB)2

16�uc�2

#

=

(ERK � ERB)2

8p2�u

!c�2

�(�� 2)�K

c12 +

�U (1� �)2p2�u

�

28

@EU�true(c)@c > 0 if

�(a�2)�K

c12 + �U (1��)2p

2�u

�> 0 for su¢ ciently small c.

Note,

limc!0

�(a� 2)�K

c12 +

�U (1� �)2p2�u

�=�U (1� �)2p

2�u> 0

limc!0

@EU�true(c)

@c= lim

c!0

(ERK � ERB)2

8p2�u

!1

c2

�(�� 2)�K

pc+

�U (1� �)2p2�u

�=1

Given that @�@c < 0, if � < 1p

3, and under consideration of the restraints

for � derived from @EU�(c)@c < 0, optimal choices for �� can be found between

1�q

23 < �

� < 1p3.

5.3 Appendix 3: Proof of Proposition 3 principal�s ad-verse selection of agents

The continuous set of competing agents is given by

agents := fj 2 N j j describes an agent with concealing cost cj 2 Rg

Proposition 2 says that @EU�(c)

@c < 0 is e¤ective. Hence, the principal will choosethe agent j 2 agents, that has the lowest conceiling costcmin = cj = c

�.

Simultaneously the same argument holds for the risk aversion �u, because of@EU�(�u)

@�u< 0, thus the principal will choose the agent j 2 agents, that has the

lowest risk aversion �minu = �uj = ��u.

Furthermore, Proposition 1 says that the agent�s choice for q looks like:

q� =

p�u��Kp2c

; b� =ERK � ERB2p2c�u�K

Therefore @q@c = �

p�u��K

2p2c1;5

< 0 is valid, and for agent j it follows:

qj = q� = qmax = q(c�):

Analogously, proposition 1 implies: b� = bmax = b(c�; ��u)since

@b

@c= �ERK � ERB

4�Kp2�uc

1;5< 0

as well as@b

@�u= �ERK � ERB

4�Kp2c�1;5u

< 0

29

and�2 = (bmax)2�2K

And last but not least Proposition 2 shows that the equality EU�mintrue = EU�true(c�)

holds because of @EU�true(c)@c > 0, which holds for a small c like cmin = cj .

�

5.4

5.5 Appendix 4: Proof of Corollary 4

If the market situation presents itself as described above, principals will choosethe agent with the lowest concealing costs cmin and highest concealing e¤ortqmax, as can be seen in the behavior of the derivatives of EU�(c) and q. Propo-sition 2 says that @EU�(c)

@c is strictly negative, @EU�(c)

@c < 0 , and therefore itfollows c� = cmin. Likewise it follows,@q@c is strictly negative,

@q@c < 0 , as is

proven in Proposition, and therefore q� = qmax.Furthermore, under these conditions, they will receive the largest share of

risky assets, and with notice to b� = bmax, proven in Proposition 3, the portfoliowill become the most risky portfolio �2 = (bmax)2�2K . �

5.6 Appendix 5: Proof of Theorem 5

Let A 6= ? be the set of destructive agents, which is not empty, and P 6= ? theset of principals in a market with asymmetric information and hidden action.In this case every destructive agent j 2 A would choose an optimal portfolio(b�j ; q

�j ) and every principal i 2 P would choose an optimal (��i ; w�f;i) such as it

is described in Proposition 1.Therefore every principal i 2 P recruits an agent j 2 A, with lowest concealingcosts cminj and highest concealing e¤ort qmaxj , such as it is shown in Proposition3. As a result of that, every principal i 2 P would choose the agent i 2 A, withthe largest share of risky assets and hence the most risky portfolio, such as itis also shown in Proposition 3. Hence, this behavior leads to an ine¢ cient andexcessive risk for every principal i 2 P , because he recruits his agent j 2 A, sothat he receives the lowest expected true utility EU�mintrue;i (Proposition 3).Since all principals make the same decision for the recruitment, this leads toan ine¢ cient and excessive risk, which is endogenously and stably chosen bythe aggregate market. According to the de�nition, this risk is described assystematic risk. �

30