en

seqatiornalomitedto mld b

d execgulatord thatof ba

s PrincEurop

much higher and sometimes in excess of 80% of shareholderequity.

g banks torisk or re

lending, such a direct intervention has a cost. The interventhe labour market for banks increases bank values and docompromise lending. Further, to the extent that there is a bdesire to intervene in the labour market for bankers, it would bedesirable if any such intervention had the effect of improvingnancial stability.

Basel III has determined, through the Capital Conservation Buffer,that banks incentives to pay out rather than retain earnings needsto be managed. Leading scholars have argued that the amountbanks paid out in share buybacks and dividends was so large as

1 Oxford-Man Institute, University of Oxford, Associate Member and NufeldCollege, University of Oxford, Associate Member. Tel.: +44 2476528098.

E-mail address: john.thanassoulis@wbs.ac.ukURL: https://sites.google.com/site/thanassoulis/

2 For discussions of these interventions please see FSB (2009), Thanassoulis (2013b)and BCBS (2010).

Journal of Banking & Finance xxx (2014) xxxxxx

Contents lists availab

Journal of Bank

journal homepage: www.equity; while non-nancial rms rarely pay this much. For somenancial institutions pay as a proportion of shareholder equity is

their capital adequacy ratio. But by encouraginsuch requirements by either avoiding lendinghttp://dx.doi.org/10.1016/j.jbankn.2014.04.0040378-4266/ 2014 Elsevier B.V. All rights reserved.

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, and macroprudential regulation. J. Bank Finance (2014), http://dx.d10.1016/j.jbankn.2014.04.004meetducingtion ines notroaderBonus Rule; and the adoption in Basel III of a Capital ConservationBufferwhich prevents banks making some remuneration paymentsif their Tier 1 capital should fall below a specied level.2 The levelof pay is indeed a signicant cost for banks. Thanassoulis (2012,Figs. 1, 3, and IA.1) documents that for a substantial minority ofnancial institutions remuneration exceeds 30% of shareholder

to assets leans against the competitive externality which drivespay up. Such a cap acts to lower aggregate remuneration. Hencebanks will have increased resilience to shocks on the value of theirassets due to their reduced cost base. This reduction in bank risk isachieved whilst increasing bank values.

In principle banks can always be made less risky by increasingBank regulationFinancial stabilityBankers payBonus capsCapital conservation buffer

1. Introduction

The remuneration of bankers ansector is the focus of signicant reEU and globally. Many are concerneof pay contributes to the riskinessinspired the Financial Stability Boardsation Practices; the adoption by theutives in the nancialy attention in the UK,the level and structurenks. This concern hasiples for Sound Compen-ean Union of the 1-to-1

This paper studies the impact of a cap on total remuneration forbankers in proportion to the risk weighted assets they control.Such a cap could be targeted, affecting some sectors, such as thewholesale side, and not others, such as the retail side. Thus thecap can work with existing regulatory attempts to treat wholesaleand retail banking separately (the Independent Commission onBanking ring-fence in the UK for example).

The analysis demonstrates that a variable pay cap in proportionG21G38

Keywords:

equivalent to more than 150 basis points of extra Tier 1 for UBS, for example. 2014 Elsevier B.V. All rights reserved.Bank pay caps, bank risk, and macroprud

John Thanassoulis Warwick Business School, University of Warwick, United Kingdom1

a r t i c l e i n f o

Article history:Received 13 June 2013Accepted 4 April 2014Available online xxxx

JEL classication:G01

a b s t r a c t

This paper studies the convalue, and bank asset allocagainst a competitive extebility of reduced lending frbank has to focus on a limlation to encourage banksSuch an intervention woutial regulation

uences of a regulatory pay cap in proportion to assets on bank risk, bankns. The cap is shown to lower banks risk and raise banks values by actingity in the labour market. The risk reduction is achieved without the possi-a Tier 1 increase. The cap encourages diversication and reduces the need anumber of asset classes. The cap can be used for Macroprudential Regu-ove resources away from wholesale banking to the retail banking sector.e targeted: in 2009 a 20% reduction in remuneration would have been

le at ScienceDirect

ing & Finance

elsevier .com/locate / jbfoi.org/

Table 1Remuneration reduction expressed as a gain in Tier 1 ratio.

10% 15% 20% 25% 30%

gatell an

2 J. Thanassoulis / Journal of Bankingto materially inhibit real economy lending through the lastnancial crisis (Acharya et al., 2009). Thanassoulis (2012) docu-ments that the banks in this study typically paid out double theamount in remuneration than they did on share buybacks and div-idends, and the shareholder payments only grew to be comparableto remuneration during the last crisis. Thus if payments to share-holders became high enough to be a concern to the well functioningof the banking system, the aggregate wage bill is at this elevatedlevel of note permanently. To determine more quantitatively thescale of the relevance of remuneration to nancial stability let ussuppose the total remuneration bill could be reduced by some per-centage. One can calculate how much of an increase in the Tier 1capital ratio this reduction in remuneration would represent bycomparing funds saved to total risk weighted assets. As remunera-tion falls during crisis periods I focus on crisis years to avoid mis-leading estimates of the importance of remuneration. Table 1considers the remuneration paid in 2008 and 2009, during the lastnancial crisis, by the top 100 global banks ranked by asset value in2011.3 If the total remuneration bill was cut by only 5%, then thiswould be equivalent to an average increase in Tier 1 equity levelsof 9 basis points. If the remuneration bill could be cut by 20% thenthe equivalent increase in the Tier 1 ratio would be 37 basis pointson average.

Table 1 demonstrates that lowering pay has only a modest effecton an average banks resilience. The average however hides widevariation amongst individual banks. Thus an intervention on paywould be targeted. It would make the banks with the most unsafepay levels, safer. Fig. 1 displays the identity of the 20 banks (inthe top 100) who would have been helped most by a 20% reductionin remuneration costs on their 2009 remuneration bill. Fig. 1 dem-onstrates that an intervention in the level of remuneration wouldhave helped some major household names which were the focusof considerable regulatory attention during the crisis. For example,a 20% reduction in the remuneration bill in 2009 would have beenequivalent to a Tier 1 increase at UBS of 1.5% (150 basis points), 1.3%for Credit Suisse, and over 0.8% for Deutsche Bank. These are signif-icant gures in the context of the Tier 1 requirements of Basel III.Thus an intervention which lowered market remuneration levelsand increased bank values would have an arguably signicant andtargeted effect of lowering risk in the nancial system.

In a market, such as the labour market for bankers services,competition to hire scarce talent leads to an externality. The mar-ket level of remuneration will be determined by the institutionwhich is the marginal bidder for the banker. By bidding to hire abanker unsuccessfully, the marginal bidding bank drives up themarket rate of pay in the nancial sector. The bidding is a pecuni-

Reduction in aggregate bank remuneration 5%

Average equivalent increase in Tier 1 levels (basis points) 9

Notes: The table expresses the money saved by a hypothetical reduction in the aggredetermining the dollar saving from a given percentage reduction in the total pay biary externality: the banker gains, the employing bank loses. How-ever, in addition the employing banks fragility to market stress isincreased by increases in its cost base. This lowers the value of theemploying bank further. This latter competitive externality repre-sents a market failure. A bank failure makes other bank failuresmore likely, and in addition can have negative consequences for

3 The data sample is the top 100 listed institutions in Bloomberg by total assets in2011 for which relevant data exists and whose activities include banking. Only groupentities were included; public institutions such as central banks and developmentbanks were excluded. Of the 100, a sample of 80 banks remain. The list includes the31 Globally Systemically Important Financial Institutions dened by FSB (2011).

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004both savers and borrowers. These further externalities magnifythe importance of the market failure.

A cap on pay in proportion to assets impacts on the marginalbidding bank more than the employing bank. As pay in a givenbusiness line rises in proportion to the resources or assets beingmanaged, in equilibrium the marginal bidding bank does not havea sufciently large pot of assets to attract the banker, and so isunwilling to offer a large enough expected payment. The bankwhich succeeds in hiring the banker will be able to do so at a lowerbonus rate as it adjusts the rate for the fact that it has a larger potof assets, and/or is an otherwise more desirable place to work. Acap on the size of remuneration in relation to assets thereforeimpacts the ability of the marginal bidder to drive up pay. Hencethe level of pay in the whole market is reduced.

As the proposed cap is on total remuneration, the measureallows the bank to structure pay in the manner it considers opti-mal. Risk sharing features, such as bonuses, can be fully preserved(Thanassoulis (2012)), as there is no requirement to force xedwages up within the cap.

A cap on pay in proportion to assets will alter a banks asset allo-cation decisions. Within an individual business unit the managerwould like to be assigned as large a fraction of the banks assetsas possible as this would likely translate into the largest pay. Thiseffect exists whether or not there is a cap, and forces banks tobecome focused on asset classes considered to be core so as tosecure the talent they desire. A cap in proportion to assets is morebinding on the marginal bidder than on the employing bank. Henceeach bank will nd that in its core business lines it is able to hire itsstaff more cheaply as the marginal bidders are impeded in theirbidding. This allows the banks to row back on the specialisationthat had been necessary with unconstrained bidding, and so bene-t from increased diversication.

The cap could naturally also be a tool for macroprudential reg-ulation as it can be used to encourage the re-targeting of banksfrom some business lines to others. Suppose that a cap is imposedon bankers managing wholesale assets, and not for those managingassets on the retail side. Those banks which were the runners-up toemploy the best wholesale bankers become less aggressive biddersdue to the pay cap. This lowers the remuneration level of whole-sale bankers and allows the banks which specialised in wholesalebanking to devote more of their assets to retail banking so as tobenet from diversication. Secondly some universal banks willbe competing against other non-bank nancial institutions whichmay be regulated under different rules. The presence of these insti-tutions outside the regulatory net strengthens the macroprudentialtool. Regulated banks would be at a disadvantage in hiring the best

19 28 37 47 56

pay bill expressed as the equivalent increase in Tier 1 equity. This is calculated byd dividing by the total risk weighted assets. Data from Bloomberg, see Footnote 3.& Finance xxx (2014) xxxxxxtraders or wholesale bankers. Hence the expected return bankswould have from these wholesale activities would decline as thebanks would be unable to hire the most sought-after traders. Thusbanks would be even more incentivised to reassign assets at themargin from wholesale towards retail banking.

2. Literature review

The objective of this paper is to investigate the consequences ofa regulatory pay cap on bank risk, bank value and bank asset allo-cation decisions. This work builds on Thanassoulis (2012) who

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

h dor 1 ranks

J. Thanassoulis / Journal of Bankingdemonstrates the competitive externality operating though thelabour market which drives up pay and so increases bank risk. Inthis study I extend the Thanassoulis (2012) framework to studythe effects of a regulatory cap on total pay in proportion to assets.Further I extend the study to consider multiple asset classes,asset allocation, and macroprudential regulation. The model of acompetitive labour market used here builds on the seminal contri-butions of Gabaix and Landier (2008) and of Edmans et al. (2009).Relative to these works I explicitly model the possibility of bank

Fig. 1. Equivalent gain in Tier 1 ratio for the 20 most affected banks. Notes: The grapreduction in the remuneration bill can be measured in terms of an increase in the Tiemost affected banks in the sample of the top 100 banks used in Table 1. These are bbeen reduced. Data from Bloomberg, see Footnote 3.failure arising from poor asset realisations, and so am in a positionto discuss bankers and their impact on nancial stability.

As in Wagner (2009), if the size of the pool of assets should fallbelow some level, a default event occurs which results in extracosts for the bank. Wagner however does not investigate the sup-ply side competition for bankers and so is silent on banker pay ingeneral. The aim of this paper is to understand how interventionin the labour market for bankers would alter bank risk.

There is little empirical evidence on the level of bankers payand on bank risk. Cheng et al. (2010) is a notable exception whichdemonstrates that nancial institutions which have a high level ofaggregate pay, controlling for their size, are riskier on a suite ofmeasures. A complementary nding is offered by Fahlenbrachand Stulz (2011) who demonstrate that bank CEOs with the largestequity compensation were more likely to lead their banks to lossesin the nancial crisis. Other empirical research has in generalfocused on CEO pay and incentives whereas our focus here is onremuneration more widely.4

This analysis focuses on the aggregate level of risk which a bankwould knowingly allow their bankers to take on rather than therisk choices of individual bankers. Other studies have focused onhow competition between banks affects the shape of the remuner-ation contracts offered, and so individual bankers incentives totake risks. For example Thanassoulis (2013a) argues that competi-tion for bankers drives pay up and can lead to an industry usingcontracts which tolerate short-termism. This work provides a

4 See for example Llense (2010) on CEO pay for performance, and Edmans andGabaix (2011) on the relative value of contract design versus hiring the optimalindividual to be CEO.

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004rationale for forced deferral of pay conditional on results. By con-trast, Foster and Young (2010) argue that any variable pay can begamed and can lead to risk being pushed into the tails. Raith(2003) considers rms competing, rather than banks, and endoge-nises the level of bonus to incentivise effort. He shows that rmswith larger market shares increase the bonus incentives they offer.Benabou and Tirole (2013) consider competing rms using con-tracts to screen workers by ability: the high ability workers aregiven incentives to take excessive risks. Acharya et al. (2013) study

cuments the impact of a 20% reduction in remuneration in the crisis year 2009. Theatio. The graph documents the impact of such a reduction in remuneration on the 20which would gain most resilience if the remuneration level of bankers could have& Finance xxx (2014) xxxxxx 3the incentives a banker has to move institution to avoid theiremployer learning whether their performance was due to skill orluck. The insights in these works are complementary to the analy-sis here as none of these analyses explore the impact of pay caps onthe labour market equilibrium.

3. The model

Suppose there are N banks who have assets in a given asset classof S1 > S2 > > SN . Banks seek a banker who will maximise theexpected returns from their assets. If the banks assets in this classshould however shrink to be less than gS, for some g < 1, then thebank incurs some extra costs. The parameter gmeasures a requiredpreservation rate on assets below which the bank, or its creditors,take actions which generate a cost to the bank. This captures, forexample, the costs of forced asset sales to reimburse creditors, orincreased costs of capital. I refer to the case in which assets fallbelow this critical level as a default event. I assume the banks costsin the case of a default event are proportional to the initial level ofassets: kS. The functional form is chosen for tractability, but it isnot a key assumption. The key assumption is that costs of a defaultevent can arise if a banker shrinks the assets they are given to man-age sufciently.

There are N bankers who can run this asset. They expect to growthe assets they manage by a factor of a1 > a2 > > aN . Thus ifbanker i is employed by bank j then the expected assets of bank jat the end of the period will be ai Sj. An expected asset growth fac-tor of ai 1 would imply that that banker i is only expected tomaintain the dollar value of the assets he or she manages. I assume

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

Table 2Proportion of remuneration received as bonus.

8). Tng, ts th

4 J. Thanassoulis / Journal of Banking & Finance xxx (2014) xxxxxxthat each bankers distribution of realised asset growth factors aretranslations of each other so that bankers differ only in their skill.Hence the density of asset growth factors delivered by banker ncan be written as fnx 1=anf x=an, where f is a density withunit expectation implying that the expectation of fn is an. Inte-grating we have the cumulative distribution of the asset growthfactor given by Fnv Fv=an. The outside option in the labourmarket for bankers will be determined endogenously to thismodel. In addition the bankers have the option of leaving thislabour market and, for example, moving to another industry orlocation. I normalise this outside option to zero. Finally bankersare assumed to be risk neutral. There is considerable evidence thatbankers may actually be risk loving (see the evidence contained inThanassoulis (2012)). However all that is required for the followinganalysis is that bankers are not too risk averse.

As the bank is an expected prot maximiser, the shape of thedistribution of asset growth outcomes generated by the banker willonly be important if the resultant asset levels are low enough totrigger a default event, leading to the extra costs described above.In any empirically relevant calibration of this model, default will bea low probability event. Hence the relevant probability will lie inthe tail of Fn. I now follow Gabaix and Landier (2008) andThanassoulis (2012) and use Extreme Value Theory to characterisethe shape of a general distribution in its left tail. I assume that theasset growth factor generated by the bankers is bounded below byzero so that banks enjoy limited liability on their investments. Inthis case the left hand tail of the distribution of asset growth fac-tors can be approximated by

Fn v G v=an c 1Extreme Value Theory would require G to be a slowly varying func-tion.5 I restrict to G > 0 being a constant. I require cP 1 so that thedistribution function takes a convex shape.

I restrict bankers to be paid in bonuses which are proportionalto the assets they control. Thanassoulis (2012, Proposition 1) dem-onstrates that banks, as modelled here, would prefer to pay fully inbonuses rather than using xed wages as well as bonuses. Bonuspay allows banks to share some of the risk of poor asset realiza-tions with the bankers. This lowers the banks expected costs from

Total compensation bands 2008

% Base salary

500 K to 1 mn 19> 1 mn 9

Notes: Data reproduced from Financial Services Authority (2010, Table 1, Annex A3.groups, and six major UK banking groups. The sample consists of 2800 staff comprisiactivities can have a material impact on their employing bank. The table demonstrateform of bonuses.the possibility of realizations which trigger a default event. Table 2presents evidence from the UK corroborating that this all bonusrestriction is a reasonable assumption, particularly for those earn-ing the largest amounts. More recent regulatory interventions havelimited bonuses and required banks to pay staff using higher xedwages.6 Table 2 shows that if banks are given the exibility theywould elect to pay staff overwhelmingly in the form of variablebonuses.

This is not an explicit model of moral hazard, though the out-come of such models is compatible with these assumptions. As

5 See Resnick (1987). A function Gv is dened as being slowly varying at zero iflimv!0G tv =G v 1 for any t > 0.

6 See Thanassoulis (2013b) for a discussion of the European 1-to-1 bonusregulation.

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004pay is delivered in the form of variable pay conditional on perfor-mance, managers are fully incentivised. The bonus rates deliveredby this model would be in excess of any bonus rates required by anexplicit model of incentives and moral hazard. Suppressing thesubscripts momentarily, if a bank with assets S hires a banker oftype a on bonus rate q then the banker expects to receive dollarremuneration of q aS. The expected asset level of the bank atthe end of the period, gross of the cost of any default event, isa1 qS. Suppose the realisation of the asset growth factor is a.There is a default event if the realisation, a1 qS < gS. Using(1), the probability of this is Fg=1 q G g=a1 q c. Hencethe expected value of the bank at the end of the period is EVwhere

E V a 1 q S kSG ga 1 q c

2

Each bank will seek to maximise this expected value. A cap on theremuneration in proportion to assets is equivalent to setting a max-imum value for the bonus rate, q.

There is a competitive labour market for bankers. Banks bidagainst each other to hire a banker to run their assets. Each bankcan offer a given banker a targeted bonus rate q which will beapplied to the realized level of assets the banker manages. Theoffers are banker specic so that more able bankers can be offeredmore generous terms. The market is assumed to result in a Walr-asian equilibrium where an individuals pay is set by the marginalbidder for their services. This can be modelled as the banks biddingfor the bankers in a simultaneous ascending auction (seeThanassoulis (2012, 2013a)).

Finally I assume that the total size of each banks balance sheetis exogenous. The assumption that balance sheets are exogenous isequivalent to an assumption that the Board of a Bank would notdecide to change their aggregate size and debt-to-equity ratio toallow an individual to be hired. Banks may well decide to altertheir asset allocation decisions within the envelope of their chosenbalance sheet size. We will explore this in detail below.7

3.1. Discussion of key assumptions

2009

% Bonus % Base salary % Bonus

81 24 7691 11 89

he FSA required this information of UK staff for seven major international bankinghe FSA estimate, 70% of Code Staff in banks operating in the UK. That is staff whoseat, given the exibility, banks would choose to deliver the vast majority of pay in theThis model of banks competing for bankers is designed to betractable and to allow the key competitive forces which determinepay levels to be clearly explained. Underlying the model are twokey assumptions. The rst is that the risk prole of the bank isdecided by the Board and not the banker, thus bonus rates donot alter the tail risk of the institution. The second is that the bankpays out remuneration to the banker, even if the banker delivers aloss on the assets managed. This section will discuss each of theseassumptions in turn.

7 It would in principle be possible for a bank to stay within its regulatory Tier 1ratios, and yet grow assets, to increase pay, by leveraging up with safe assets. This canbe managed here by appropriate risk weighting (Section 5.1). Further this moregeneral weakness in the regulatory regime is already being addressed through theLeverage Ratio requirement in the Basel III framework.

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

The Board of any bank will determine a desired risk prole fortheir institution depending upon the return on equity they believetheir investors demand. The Board will seek to impose this riskprole on the bank by using the corporate governance levers attheir disposal. These levers include the ability to manage the Value

weighed by the extra costs incurred in remuneration to the banker.A bank recruiting for a larger set of assets would have the skill of

J. Thanassoulis / Journal of Bankingat Risk (VaR) of individual bankers, often on a daily basis, andbroader asset allocation and hedging decisions. This study assumesthat these levers are sufcient to restrict the bankers to the desiredrisk prole. Banker skill is therefore solely expressed as theexpected return given this shape of (tail) risk. If this risk controlassumption is violated then payment levels and bonus rates arerelated to the risk prole of the institution. That is the tail risk Fnwould be a function either of the bonus rate q, or of the expecteddollar remuneration.8 The dependence of tail risk on the remunera-tion would complicate the analysis offered here. When bidding tohire a banker a large bank would be able to offer a low bonus ratewhich, in the case of poor risk control, would lower the riskinessof the bank. However larger banks will secure the services of moretalented bankers who have to be paid more, and this might raisethe riskiness of the institutions. The effect of poor risk control wouldtherefore be ambiguous for the solution of the model, even absentany bonus caps. However the externalities described in this paperwould remain: the marginal bidder for a banker would increasethe fragility of the employing bank by raising her costs. This effectwould be exacerbated for larger banks if tail risk grows in remuner-ation levels, or may be mitigated if tail risk responds to bonus rates.

This study considers the impact of a cap on pay in proportion tothe assets a banker manages, and this cap is expressed as a cap onthe bonus rate payable. The intervention of a bonus rate cap stud-ied here lowers bonus rates and overall pay levels. Therefore ifbanks cannot fully control their tail risk then such an interventionwould mitigate the adverse effects of the poor risk control (seeFootnote 8). The lower bonus rates would reduce the incentive totake excessive risk, to conduct fraud, to be myopic and to churnacross employers. Hence the analysis here understates the benetsof a bonus cap along all these avenues.

The second key assumption is that even if a bank should see itsassets shrink enough to trigger the costs of a default event through,for example, forced asset sales, then the bank incurs a remunera-tion payment nonetheless. It might seem more realistic that abanker who returns a lower level of assets than she began withwould not only not receive a bonus, but most likely lose her job.If so then one might conclude that remuneration payments wouldnot add to a banks fragility, as when assets shrunk remunerationpayments would automatically be suspended until the threat of adefault event had passed. This reasoning is incomplete for a num-ber of a reasons. Firstly, it may be that a banker who shrinks assetsloses her job, however consider the following thought experiment.A banker running assets of 100 makes a 20% loss in the rst twoquarters and so is dismissed. The bank will need an alternativebanker to run these assets, suppose this replacement banker deliv-ers 10% growth in the remaining two quarters. This second bankerwould expect to be paid, and yet over the year assets have shrunkfrom 100 down to 100 80% 110% 88, a reduction of 12%.Thus remuneration is payable even if one believes that in bankingno failure is tolerated. Secondly, in reality a reduction in asset lev-els may well be due to bad luck and wider economic forces, ratherthan poor banker skill. Indeed bankers would invariably argue this

8 This may be because high bonuses are used to separate high ability from lowability bankers with the former incentivised to take excessive risks (Benabou andTirole, 2013; Bannier et al., 2013); or it may be because bankers game bonus schemesthrough legitimate and illegitimate schemes (Foster and Young (2010)); or it may bebecause bonuses provide an incentive for bankers to take early risks and then jump to

a new employer before their ability is revealed (Acharya et al. (2013)); or it may bebecause bonuses encourage bankers to push risks into the future so inducing myopia(Thanassoulis (2013a)).

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004the better banker applied to a larger pot of assets. In addition,increases in banker skill raise the expected asset growth and solower the probability of a default event. As default costs areincreasing in the size of the assets managed, the reduction in theexpected costs from default is more substantial for the bankrecruiting for a large pot of assets. Hence, for both reasons, the lar-ger bank would value the better banker more, and so the bankrecruiting for the larger pot of assets would win in bidding for agiven banker. It follows, by induction, that there will be positiveassortative matching with bankers being assigned in equilibriumto banks according to their rank.

In this benchmark case bankers are indifferent to the identity oftheir employing bank and select their employer based on theirexpected pay. The analysis offered here is essentially unchangedif banks differ in non-nancial ways. For example banks may notall offer an equally pleasant work environment, or banks may notall offer equally compelling long-term career prospects. Supposethat if a banker works at bank i, then bank specic differences raisethe utility generated for the banker by a factor of si. Thus if thebonus rate were q then the bankers expected utility at bank iwould be 1 si qaSi. In this case it is as if the banker were man-aging utility adjusted assets of Ri 1 si Si. The banks could bere-ordered according to Rif g. We would then have positive assor-tative matching by utility adjusted asset size. The results in thispaper would be unaffected by this change.to be the case. Thanassoulis (2012, Figure 2) demonstrates thatbankers were paid very large sums on average in the recent past,even after delivering negative returns on equity. Finally, unlessthe bank formally enters bankruptcy protection, remunerationcontracts have to be honoured. A bank may also wish to honourimplicit rather than explicit commitments as any failure to do sowould alter all employees expectations of their pay and lead eitherto demands to make implicit commitments explicit in contractterms, or lead to the departure of staff. Thus I conclude that theassumption that remuneration is payable even if a bank incursthe costs of a default event is appropriate.

4. The no intervention benchmark

The level of pay a banker enjoys in the market is set by the mar-ginal bidder for their services. A bank, in deciding howmuch to bidfor a banker, trades off the cost of employing the banker as againstthe increase in value the banker generates, net of any changes tothe expected costs of a default event, as compared to the next besthire. This section will determine the market rate of pay as a func-tion of fundamentals.

Lemma 1. The bank with the nth largest assets to be managed willhire the banker of the same rank n. Thus there will be positiveassortative matching.

The lemma follows by showing that a bank recruiting a man-ager to manage a large pot of assets would be willing to outbid abank which is recruiting a manager to oversee a smaller pot ofassets. This is not immediate as we are in a setting of non-transfer-able utility. Greater pay for a banker increases the expected costs ofdefault. This loss of value to the bank is not a gain to the banker.The bank recruiting for the smaller set of assets will bid for theirrst choice of banker up to the point where the extra value gener-ated on their assets as compared to the next best banker is just out-

& Finance xxx (2014) xxxxxx 5It follows that the marginal bidder for a banker of rank n is thebank of rank n 1. We are therefore in a position to solve for themarket rate of remuneration for all of the bankers:

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

and so allows the employing bank to hire the banker they woulddo absent the cap, but at a lower level of remuneration. Hencethe market rate of pay is reduced. This reduction in pay increasesthe value of the bank directly as they secure their equilibriumemployee more cheaply. In addition the reduction in the remuner-ation payable lowers the banks fragility as less remuneration mustbe paid out when the bankers realized results are poor. This reduc-tion in risk also raises the value of the bank.

As the employing bank now secures greater value from thebanker they hire, in equilibrium, to run their business unit, the sur-plus the bank is willing to bid to hire marginally better bankers is

ing & Finance xxx (2014) xxxxxxProposition 2. The banker of rank i will be employed by bank i andwill receive an expected payment of qi aiSi where the bonus rate qi isgiven by:

qi XNji1

SjSi

aj1 aj

ai3

Proposition 2 follows by an inductive argument. The amountbank i needs to pay to secure the banker of rank i depends uponhow much bank i 1, one down in the size league table, is willingto bid. This is the marginal bid which needs to be matched. Theamount bank i 1 is willing to bid depends upon how much banki 1 must pay for its banker, which in turn depends upon the bid-ding of bank i 2. Hence the market rate can be established byinduction.

Having established the market rates of pay through Proposition2 we can now interrogate the impact of regulatory interventions onthe entire market.

5. Effect of a pay cap in proportion to (risk weighted) assets

Let us now consider a policy intervention which caps the pay ofthe individual running this asset class to no more than a proportionv of assets. As I have assumed good corporate governance of bankrisk, the optimal bank risk prole which maximises returns isunchanged. So the Extreme Value approximation (1) continues tohold. Analysis of the new market equilibrium yields that such aregulatory intervention would have the following effects.

Proposition 3. Consider a mandatory cap on the remuneration of thebanker equal to at most a bonus rate v as a proportion of assets.

1. The intervention lowers bank risk and raises bank values for allexcept the smallest banks.

2. The lower the remuneration cap as a proportion of assets, thegreater the positive impact: higher bank values and lower bank risk.

3. The equilibrium allocation of bankers to banks is not affected, pre-serving allocative efciency.

In the labour market, banks compete with each other to hirescarce talent. The market rate of pay for a banker will be deter-mined by the institution which is the marginal bidder for the bank-ers services. By bidding to hire a banker unsuccessfully, poachingbanks drive up the market rate. The bidding is a pecuniary exter-nality: the banker gains while the employing bank loses. However,there is also an increase to the employing banks fragility to stress,due to increases in its cost base. The larger cost base due to payincreases the probability of a destruction of assets beyond therequired preservation level, and so increases the expected cost ofthis event. This lowers the value of the employing bank furtherand is a competitive externality. The cap works by leaning againstthis competitive externality.

The cap impacts the marginal bidder for any given banker morethan the equilibrium employer. The remuneration enjoyed by abanker is set by the amount the marginal bidding bank is preparedto offer. Lemma 1 demonstrated that a larger bank would be will-ing to bid most, yielding positive assortative matching. It followsthat the bank which succeeded in hiring a banker in equilibriumwill have been able to do so at a lower rate as a proportion ofthe assets the banker will run. The preferred bank adjusts the rateit offers down for the fact that it offers the banker more resourcesand opportunities to make prots, and/or is a more desirable place

6 J. Thanassoulis / Journal of Bankto work.A cap on pay in proportion to assets impacts the ability of the

marginal bidder to drive up pay. This lowers the marginal bid

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004reduced. The reduction in the competitive externality, and the cor-responding reduction in bank risk therefore propagates upwardsthrough the labour market.

It follows from the logic of the intervention that the moresevere the cap, the greater the impact on the marginal bidder,and so the greater is the gain for bank values, and the greater thereduction in bank risk.

As the cap applies to all banks in proportion to assets, it doesnot alter the matching of bankers to banks. No allocative inef-ciency is introduced into the system. However, the benet requiresmacro not micro prudential regulation. No single entity can securethe risk reduction and value increasing benets alone, as thesearise from altering the value of the competing remuneration offersfor any given banker.

The remuneration cap will lower market rates of pay for bank-ers. In principle one might therefore be concerned that this willlead to a departure of workers from nance to other industries.However education-adjusted wages enjoyed by workers in nancehave out-stripped other industries since 1990 by a premium ofbetween 50% and 250% for the highest paid employees (Philipponand Reshef (2012)). Thus I conclude that wages in nance could fallby some margin before the general equilibrium labour re-alloca-tion effect would become a problem.

Salary caps have been a feature of sports remuneration in theUS. However these are different to the proposal outlined here.Sports salary caps are the same across all teams,9 while the inter-vention studied here links pay caps to the size of the assets managed.This link to bank size is critical in ensuring the cap targets the neg-ative externality created by the marginal bidder, and ensuring thatthe cap does not create a distortion in the allocation of talentedbankers to banks.

Finally, it has been noted that the nancial sector has under-gone a period of sustained consolidation and merger activity datingback to before the 1990s.10 This consolidation in the banking sectorhas been accompanied by a sharp increase in the size of the balancesheets of the largest banks (Morrison and Wilhelm (2008)). Themodel of the banking labour market we study here captures one rea-son bank mergers create value: the desire to grow the balance sheetto allow more talented bankers to be hired. The pay cap studied heredoes not necessarily strengthen this merger incentive. Whether itdoes so depends on the particular parameter values.11

9 See for example 2013 NFL salary cap breakdown by team, USA Today, availableat http://www.usatoday.com/picture-gallery/sports/n/2013/09/13/2013-n-salary-cap-breakdown-by-team/2808245/.10 For example, Bank for International Settlements (2001, Table 1.1, p34) documentthat in 1990 there were 8 M&A deals involving banks in one of the 13 countriesstudied with a value in excess of $1 bn, and the average value of these deals was$26.5 bn. Over the decade this activity grew, and by 1998 there were 58 M&A deals inthat year with a value in excess of $1 bn, and the average value of these deals hadrisen to $431 bn.11 For an example of merger becoming less protable with a bonus cap consider aduopoly of banks. Merger (to monopoly) will have the merged bank hiring the best

banker and offering a bonus at the normalised rate of 0. This is unaffected by a bonuscap. Pre-merger a bonus cap can increase the value of the larger bank, hence loweringthe incentive to merge.

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

the pay cap should grow in that securitys systematic risk. Rochet(1992) argues that risk weights in capital adequacy requirementsshould be proportional to the expected returns to ensure that the

J. Thanassoulis / Journal of Banking & Finance xxx (2014) xxxxxx 75.1. Assets to be valued on a risk weighted basis

The analysis has explored the case of good corporate gover-nance under which the risk prole of the bank is set to maximisethe banks value. To ensure the robustness of the regulatory paycap, I now consider how a banker would seek to distort the valuemaximising risk prole of a bank if their objective was to maximisethe money available for remuneration.

In this Section I will use the PyleHartJaffee approach tomodel-ling the bank as a portfolio manager.12 Formally suppose a bankwishes to maximise the value generated from m securities with thereturns on security j 2 1; . . . ;mf gdenoted ~rj

. If the bank selects allo-

cations in dollars of xj

then next periods assets will be eS Pjxj~rj.These returns are assumed jointly normally distributed with vectorof expected returns q and the variance-covariance matrix V. Hence

l E eS Xj

xjqj

r2 var eS x;Vxh iThe PyleHartJaffee approach assumes that the value function ofthe bank can be decomposed into a function of only the rst twomoments of the returns distribution: U l;r2

. If the bank selects

the riskiness of her portfolio, as assumed here, then the rst ordercondition of the banks optimisation problem would yield

@U@l

@l@xi

@U@r2

@r2

@xi 0

This can be written in matrix notation as kq Vx 0 wherek @U=@l =2 @U=@r2 . Hence the bank would select an alloca-tion of assets for the banker to manage proportional to V1q.

I now assume that the banker managing these assets must bepaid an amount W for past performance. Suppose that any capon remuneration applies to the weighted sum of security valuesb; xD E

with vector of weights b. Thus the pay cap regulation implies

W 6 v b; xD E

To analyse the scope for banker induced distortion, suppose that thebanker can distort the risk prole of the bank, as long as he deliversa value of the objective U l;r2

of at least R. As the banker wishes

to maximise his pay, his optimisation problem becomes

maxx1 ;...;xmf g

v b; xD E

subject to R U x;qD E

; x;Vxh i

4

Proposition 4. The ratio of allocations to individual securities isunaffected by a pay cap if the cap weights securities proportionally totheir expected returns (b parallel to the vector of expected returns q).

The banker will be tempted to alter the investment prole hetargets if doing so can allow more to be paid under the cap whilstpreserving the expected returns net of risk. Proposition 4 showsthat this is not possible if the weights used to measure the quantityof assets are proportional to the expected returns on those assets.Hence if assets are weighted proportionally to expected returns,the assumptions of this analysis remain robust, even if the bankerselects his investment strategy so as to maximise pay.

This analysis parallels that underlying the derivation of optimalrisk weights in capital adequacy regulation (Rochet (1992)). In thestandard CAPM framework, the expected return on a securityrewards the investor for the securitys undiversiable risk. HenceProposition 4 captures that the weight accorded to a security in12 The PyleHartJaffee approach was proposed in Pyle (1971) and Hart and Jaffee(1974). The version used here is derived from Freixas and Rochet (2008, Section 8.4).

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004bank will invest in an efcient portfolio of assets given the limitedliability constraint. To the extent that the Basel risk weights capturesystematic risk, they are a convenient approximation to this rule.13

6. Asset allocation responses to a pay cap

Banks invest in many asset classes. The banker managing anasset class can make greater prots from a larger pot of assets.Hence, even absent pay regulation, there exists an incentive totry to manage as many assets as possible. This implies that in theabsence of any remuneration cap banks are under pressure to raiseasset allocations to areas where they seek to hire the best bankers/traders. This increased asset allocation has a cost however in termsof reduced diversication and excessive concentration.

This section will demonstrate that a remuneration cap does notstrengthen this effect, but rather weakens this excessive concen-tration effect amongst evenly matched banks, and so creates anincentive for banks to re-assign assets so as to better diversify.The cap impacts the marginal bidder in any asset-class more thanthe equilibrium employer. It therefore hampers the extent towhich a rival bank can drive up remuneration in any given assetclass. This reduces the need to focus assets on a limited numberof core areas, and so allows for greater gains from diversication.

I demonstrate these results through an extension of the modelto allow for multiple asset classes.

6.1. Extension to a model of multiple assets

The diversication effect of bonus caps is at its strongest whenthe competing banks are close in size. Later in this Section I will dis-cuss the case of banks of very different size. To demonstrate thispositive effect of bonus caps most simply, consider initially twobanks eachwith equal total balance sheet size of T. Consider amodelof two available asset classes, and within each asset class there aretwobankerswhocould runeitherbanks allocation to the asset class.The most able manager in each class has an expected growth factorof a, the next best hire has an expected growth factor of b < a. Thebankers outsideoptions continue tobenormalised tozero. Theassetlevel realisations in each asset class are assumed to be independent.

Each bank must decide how to split its balance sheet betweenthe available asset classes, assigning S dollars to one asset classand T S dollars to the other. To proxy for the benets of diversi-cation parsimoniously I suppose that the banks gain valuec S T S on top of the assets realised within each asset class,with the parameter c a constant greater than zero. The specicfunctional form of diversication benet is for convenience, theeconomic assumption is that diversication confers some benetsto the bank, and these benets fall away if the bank withdrawsfrom a given asset class. This assumption captures, for example,that the volatility of returns in the normal course of business arereduced which provides value for employee stock holders andany other investors who are not fully diversied; alternativelythe assumption captures the effect of diminishing marginal returnsto any given asset class as more and more of the balance sheet isused for that asset class. I model the banks as rst simultaneouslydeciding their asset class allocations, and then competing to hirethe bankers as in the benchmark model given above.

13 However the risk weights offered in banking regulation are not a pure estimationof systematic risk (Iannotta and Pennacchi (2012)). The Basel rules allow nationalregulators some exibility in selecting risk weights, and where analysis is conducted

the risk weights are calculated to reect the overall expected loss conditional on adefault. The Basel risk weights will therefore be a good proxy for systematic risk onlyto the extent that systematic risk is correlated with overall risk.

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

6.2. Optimal asset allocation

a default event is reduced. Hence each bank responds to a relaxa-

in this case exactly parallels the single asset class analysis in Section 5.

and two bankers in each asset class with expected asset growth

its asset allocation choice. The bonus rate paid by bank w for thewholesale banker will be below the cap (13). There is no cap on

8 J. Thanassoulis / Journal of Banking & Finance xxx (2014) xxxxxxtion of the pay cap by focusing more on its target asset class indefence against the now more aggressive rival bank.

Running the process in reverse we see that as the remunerationcap becomes more severe, it is the institutions which are alreadymost devoted to the class that are least handicapped. The cap ismore binding on the marginal bidder in each asset class than onthe equilibrium employer. It therefore follows that the leadinginstitutions in the class are in a position to reduce their asset allo-cation as they can continue to employ the best staff with fewerassets, and stand to gain the diversication benets by re-balanc-ing towards other asset classes.

Hence an effect of the pay cap intervention is that it reduces thepressure for similarly matched banks to excessively focus on theircore areas, as would be necessary with unconstrained bidding.The cap instead creates a force for diversication amongst thebanks. This benecial effect becomes weaker as the banks becomemore asymmetric in size. To see this suppose that the two banksstudied in this section become sufciently asymmetric in size thatthe large bank can secure the a-banker in both asset classes. In thiscase one can show that the optimal asset allocation will be unaf-fected by the presence, or otherwise, of a bonus cap.14 The analysis

14 Both banks would split their balance sheets equally between the asset classes tomaximise the diversication benets. If the bonus cap is binding, then the smallerBecause of the symmetry of this initial problem I consider asymmetrical allocation. I therefore consider an allocation of assetsin which each bank targets the best banker in a different asset classby putting S > T=2 into the targeted asset class, and T S into theremainder. The expected value of a bank which secures thea-banker in its targeted class for a bonus of qa, and the b-bankerin the other business line for a bonus of qb is given by

V S; T S a 1 qa S kSGg

a 1 qa c

c S T S

b 1 qb

T S k T S G gb 1 qb !c 5

Eq. (5) captures the costs of a default event in any asset class. Suchan event occurs if the assets under management in the class shrinkto be less than g of their initial level. Under a pay cap we requireqa; qb < v.

Proposition 5. As the cap on pay becomes more severe (v declines),banks re-balance their asset allocation in the direction of making theirexposure more diversied and less asymmetric.

The asset allocation a bank makes is a trade off between givingthe most assets to managers who can produce the highest return,set against the costs of over-specialisation. To understand theresult it is perhaps easiest to consider the reverse, and suppose thata remuneration cap becomes less binding. As the remuneration capis removed, each bank nds itself subject to more aggressive bid-ding for the best banker from the bank which is under-weight inthat asset class. To continue to employ the a-banker in its targetedasset class, each bank must match the more aggressive bidding.This lowers the prots available from the asset class, and itincreases the risk of a default event as well. If the bank nowincreases its asset allocation to its targeted area then it can lowerthe proportion of the realised assets used for remuneration. Thisincreases the banks value from this asset class because its risk ofbank will bid at most a bonus rate of v. The larger bank would secure the a-banker ineach asset class at a bonus rate of vT2=T1 where T1 > T2 denotes the size of the totalbalance sheet. The equilibrium bonus falls in the bonus cap as per Proposition 3.

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004bonuses offered to bankers in the retail banking asset class.I again model the banks as rst simultaneously deciding their

asset class allocations, and then competing to hire the bankers asfactors a > b. For expositional purposes, and in keeping with themotivating example, I will label the two asset classes r for retailand w for wholesale banking. However the analysis applies toany subdivision of banks activities. Generalising from Section 6, Imove away from symmetry and consider two banks with balancesheets Tr ; Tw. I restrict attention to the interesting case in whicheach bank secures just one of the a-bankers. Bank Tr will specialisein the r asset class (e.g. retail banking). It devotes Sr dollars to retailbanking, and Tr Sr dollars to the alternative asset-class: whole-sale banking. Similarly bank Tw specialises in the w asset-class(e.g. wholesale banking), and so devotes Sw dollars to its asset classspecialism (the w asset class). Bank Tr assigns more dollars to the rasset class than the rival bank, and in this sense specialises in the rasset class (retail banking).

Each bank secures gains from diversication (as in Section 6)proxied by c Sw Tw Sw for the wholesale focused bank, and sim-ilarly for the retail focused bank.

The regulatory intervention I analyse here is a bonus rate cap vapplied to remuneration on the w-asset class only. If this bonus capis binding then itwill affect themarginal bidding bank in thew-assetclass. Hence the cap implies that bank r is restricted in the bonus rateit can offer to try and attract the a-wholesale banker. Bank r can offerthe a-wholesale banker at most expected pay of v a Tr Sr givenA bonus cap impacts the ability of the marginal bidder to drive upremuneration in both asset classes, allowing the larger bank to lowerits risk and increase its value with no asset allocation distortion.

7. Pay Regulation for macroprudential objectives

A cap on remuneration in proportion to assets can be applied tosome business lines and not to others. This section demonstrateshow such partial application of pay regulation can be used to re-target banks activities to certain asset classes. Suppose, as anexample, that for reasons outside of this model a regulator decidedthat there was insufcient lending to the real economy via banks.15

In this case a pay cap in proportion to assets applied to bankersworking in wholesale banking, but not in retail banking, would alterthe equilibrium asset allocation decisions so that all banks refocusassets away from wholesale and towards retail banking. Though apay cap is an instance of microprudential regulation, the effectwould be a macroprudential one as the resilience of all banks acrossthe system is improved.

Further banks are in competition with other Financial Institu-tions, such as hedge funds, to secure bankers/traders, and thesenancial institutions who do not possess a banking license areoften regulated under different rules. I will study the case ofincomplete regulatory coverage in Section 7.2. The existence ofnancial institutions outside the regulatory net, rather than beinga problem, can be used to further enhance the efcacy of pay-capsas a macroprudential tool.

7.1. A model of partially applied pay cap regulation

Once again consider the model of Section 6 of two asset classes15 Insufcient lending in the UK to Small and Medium sized Enterprises (SMEs) hasbeen a notable recent regulatory concern. See for example Funding for Lendingfailure dismays BoE, Financial Times, March 11, 2013.

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

from the pay cap. (Otherwise the analysis above is triviallyextended). This model simply captures that banks have multiplebusiness units, and in some of the business units they will face riv-als who come under a different regulatory regime. The benets ofdiversication for bank r are again proxied by c SrTr Sr if bank rassigns Sr dollars to the retail banking book. In both asset classesthere continue to be two bankers with expected growth factorsa > b. (Only one retail banker will be required hence the a-retailbanker will be secured by bank r.)

The case of interest is where, absent any cap, the bank wouldsecure the better executive to run its wholesale business unit. Sucha bank is one which is vulnerable to the introduction of a remuner-ation cap which applies to it, but not its rival. To this end I restrictattention to the case in which

Sh max1Sw

;1Sr

k2G

ga

c c c 1 v21 v c2

6

This assumption delivers stability of the equilibrium allocation ofassets between classes. The assumption is trivially satised if thebanks are large enough.

Wearenow inaposition to study theeffect apartially appliedpaycap has on the asset allocation decisions of the two banks. I denotethe best asset allocation response of bank w to bank r as Sw Sr andvice-versa. Thus if bank r assigns assets Sr to its specialism (the rasset-class, retail banking), and by implication assets Tr Sr towholesale banking, then bank ws best response is to assign Sw Sr to its asset-class specialism (the w asset class, wholesale banking).

Lemma 6. The best asset allocation responses of each bank arestrategic substitutes. Thus dSw Sr =dSr < 0.

The result builds on the logic of Section 6 and demonstrates thestrategic interaction between asset class allocation decisions. Ifbank r should increase its allocation to its asset-class specialism(r asset-class, retail banking), then by denition it is moving assetsaway from the other asset class: wholesale banking. The rival banknow faces a less aggressive bidder for the a-wholesale banker. Asexplained in Section 6 the wholesale focused bank can now benetfrom increased diversication and so reduces its focus to wholesalebanking. The wholesale focused bank therefore increases its alloca-tion to retail banking. As there is no bonus rate cap on remunera-tion to retail bankers, there is now a second round effect makingbank w a more aggressive bidder for the a-retail banker. Therefore,to protect its protability bank r optimally responds by furtherincreasing its asset allocation to retail banking also.

Bonus caps applied to wholesale banking can kick-start this re-allocation process by inhibiting bank r from bidding up wholesalebanker bonuses:

Proposition 7. If a bonus cap applying only to one asset class is mademore severe, all banks increase their asset allocation to the alternativeasset class. Hence if a bonus rate cap v applying only to wholesalebanker remuneration is reduced, all banks increase their asset alloca-tion to retail banking.

A bonus rate cap applied to one asset-class affects the marginalbidders ability to drive up pay in this asset class. The retail-focusedbank is the smaller bank in the wholesale banking asset class, andso it is the marginal bidder setting the remuneration level whichthe wholesale-focused bank needs to match. A bonus rate cap forbankers working in wholesale banking impedes the retail focusedbank from bidding up the remuneration of wholesale bankers. Thissets off the logic of Lemma 6. Namely the wholesale focused bank,facing less intense competition to hire the a-wholesale banker, isable to prot from diversication. Thus bank w, at the margin,moves some assets away from wholesale and towards retail bank-ing. This makes competition for the a-retail banker more intense,and as there is no bonus cap to protect it, this leads to the retailfocused bank also repatriating some of its assets away from whole-sale and towards retail banking. This effect is depicted graphicallyin Fig. 2. Thus partially applied pay caps can be used to alter banksasset allocation decisions through the economic cycle.

7.2. Macroprudential effects with incomplete regulatory coverage

J. Thanassoulis / Journal of BankIn this Section I expand the analysis above to demonstrate why,even with incomplete regulatory coverage, pay caps in proportionto assets applied partially across asset classes, can be used effec-

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004tively to alter the equilibrium allocation of banks assets. Considertherefore just one universal bank r active in both the r asset class(e.g. retail banking) and the w asset class (e.g. wholesale banking).The bank is once again regulated as to the remuneration it can payto bankers who manage assets within its wholesale banking book.It is not regulated on payments to those managing retail bankingassets. Thus the pay cap regulation continues to be partially applied.

Now replace the universal bank w analysed in the section abovewith a competing nancial institution active only in the w assetclass. I will refer to this institution, for the purposes of this exam-ple, as a hedge fund and label its assets in the class Sh. I assume thish institution sits outside of the regulatory net and so is exempt

Fig. 2. Best response asset allocation functions. Notes: The curve SwSr captures thebest asset allocation response of the w focused bank on asset class w (wholesalebanking), in response to the allocation Sr of the r focused bank to the r asset class(retail banking). The best response functions are strategic substitutes as the curvesslope down. As the bonus rate cap v on wholesale banker remuneration is mademore severe (v declines), the best response curve of the w bank is pulled down. Thebest response curve of the r bank is not affected as there is no cap on retail bankerremuneration. Hence the equilibrium asset allocation to retail banking rises forboth banks.

& Finance xxx (2014) xxxxxx 9Syr Tr2 12c

a b kG gb

ga

c

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

atively stronger in banking than in other industries. Secondly, thenancial sector has a larger remuneration bill as a proportion ofshareholder equity than other industries. It therefore follows thatthe gain from a pay cap in terms of bank risk reduction is corre-spondingly greater than it would be in other industries.

Acknowledgements

I would like to thank the editor Ike Mathur, and an anonymousreferee for detailed comments which have greatly improved thispaper. I would further like to thank Sam Harrington, Su-Lian Ho,Victoria Saporta, Matthew Willison, and the other members ofthe Prudential Policy Division at the Bank of England for helpfuldiscussions. I would also like to thank the Bank of England for their

ingThe economics of Lemma 8 are readily explained. Suppose, for acontradiction, that the bank only succeeds in hiring the b-banker torun the wholesale banking book. Under this assumption the valueof the bank can be determined by adapting (5) as:

Wr Sr aSr kSrG ga c

|{z}from retail book

b Tr Sr k Tr Sr G gb c

c Sr Tr Sr 8Eq. (8) follows as both bankers will receive a normalised bonus rateof zero. This value function is concave in the allocation of assets tothe retail banking book, Sr . Hence there is an optimal allocationgiven by the rst order condition. This asset allocation is sufcientlylarge that the bank would have more assets in its wholesale bankingbook than the hedge fund, given assumption (7). This delivers thedesired contradiction as the bank will outbid the hedge fund andso secure the a-banker for its wholesale activities (Lemma 1).

It follows that, absent pay cap regulation, bank r will outbid thehedge fund and hire the a-wholesale banker. As bank r must com-pete with the hedge fund to secure the wholesale banker, the a-wholesale banker receives higher remuneration than the a-retailbanker does. Thus, to protect its protability the retail bank divertsassets to wholesale banking, shrinking its retail banking book. Thisis not straightforward to show as the interaction between pay lev-els and bank default risk is not linear. Nevertheless it can be dem-onstrated that we have an upper bound on the retail banking bookin the absence of pay cap regulation, and this upper bound is givenin Lemma 8.

Proposition 9. If the bank is subject to a sufciently severe cap onremuneration for the wholesale banking book then the bank will re-allocate more assets to retail banking and reduce the size of itswholesale banking book.

Proposition 9 considers a regulation which is sufciently severethat the bank loses the best wholesale banker to the hedge fund. Inthis setting the bank can secure bankers, but in wholesale bankingthey are not the very best ones. As a result the expected growthfactor available from wholesale banking assets falls slightly, tothe lower level of b. The bank would now conduct its asset alloca-tion decision as in the proof of Lemma 8 under the assumption thatit will secure the b-bankers for the wholesale banking book, and sothe optimal asset allocation can be found. At the asset allocationstage the bank will choose, at the margin, to divert funds awayfrom the wholesale banking book and towards the retail bankingbook as the returns from wholesale banking have diminished asa result of the partially applied pay cap regulation. Proposition 9captures that the incomplete regulatory coverage of remunerationregulation can be turned to the regulators advantage. The ability touse pay cap regulation as a macroprudential tool survives in thepresence of a porous regulatory net.

8. Conclusion

A variable cap on remuneration in proportion to risk weightedassets lowers bank risk and raises bank values. Such a cap impactson the marginal bidder for a banker more than on the employingbank. The implication is that the market rate of pay for bankersdeclines, and so banks become less fragile as their cost base ispulled down. By addressing a negative externality in the labourmarket for bankers, the intervention also has the effect of dampen-ing the pressure banks are under to focus resources on given asset

10 J. Thanassoulis / Journal of Bankclasses so as to secure better bankers. And the pay cap can be usedto achievemacroprudential objectives through the cycle as it can bestructured to encourage banks to refocus towards a subset of asset

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004classes (e.g. retail banking) if desired by a regulator. Finally, byusing appropriate risk weights, bankers incentives to abuse anyweakness in corporate governance failings to grow pay is mitigated.

Consider therefore a regulatory intervention which capped totalbank remuneration summed over wholesale bankers proportionalto each banks risk-weighted wholesale banking assets. Regulationat the aggregate level is easier and less costly to implement than perperson caps. And yet such a cap will likely be implemented byseniormanagement on rank-and-le hiring decisions as a top downrule. This is because the numbers of employees involved wouldmake micro-managing deviations from a general rule impractical(see Table 3). Hence a cap at the bank level tackles the externalitydescribed at the individual banker level, and likely generates theconsequences for bank values and bank risk studied here.

As a benchmark calculation let us suppose that remuneration inbanks adhered to a commonly experienced 80:20 rule (Sanders(1988)) so that the 20% best paid bankers secure 80% of the remu-neration. If the pay of these best paid executives could be loweredby a quarter then this would equate to a 20% reduction in the over-all remuneration bill, the effect of which was graphed in Fig. 1.Such a reduction in 2009 would have been equivalent, in safetyterms, to an increase in the Tier 1 ratio of over 150 basis pointsfor the most affected institution (UBS).

The logic, described in this analysis, of the negative externalitybanks exert on each other through the labour market exists in allindustries. Thus one might wonder if a similar pay cap regulationwould be advisable in other industries beyond nance. I do notseek to take a stand on this question. However I note that the ratio-nale for intervening beyond nance is weaker for at least two rea-sons. Firstly the nance industry is special as compared to otherareas of business due to the negative externalities it exposes soci-ety to when nancial rms fail. These impacts on society are notformally part of this model and so this study does not offer a jus-tication that pay caps in banking are worthwhile. I purely notethat the case for pay caps in proportion to assets is likely to be rel-

Table 3Numbers of employees targeted by intervention on top 20% of earners.

20% Of employees in 2009

UBS 13,047Credit Suisse 9520Morgan Stanley 12,278Deutsche Bank 15,411Goldman Sachs 6500Citigroup 53,060

Notes: The table documents the numbers of employees which would have to becaptured by an intervention if it were targeted at the top 20% of earners in thenamed banks in 2009. The data is drawn from Bloomberg and the dataset is thatused in Table 1 and Fig. 1. The banks displayed are a selection of household namesdrawn from the top 20 banks documented in Fig. 1.

& Finance xxx (2014) xxxxxxhospitality whilst I was undertaking this research. Finally I amgrateful to audiences at the Financial Globalization and SustainableFinance Conference, Cape Town, the IFS School of Finance, the CFA

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

rate qi;j then the bonus must satisfy ajqi;jSi u. Hence bank is

Setting (9) equal to (10), this has solution aj1 1 qi;j1

ji1

(14). This moves monotonically with v delivering the result.

ingaj 1 qi;j

. The maximum bid that bank i will make for bankerj 1 is thereforeqi;j1 1 aj=aj1

1 qi;j 11

The same working determines the maximum that bank i 1 is will-ing to bid for banker j 1 as qi1;j1 1 aj=aj1

1 u= ajSi1

.

The lemma follows by demonstrating that bank i 1 is willing tobid to higher levels of utility for banker j 1:aj1Si1qi1;j1 aj1Siqi;j1 aj1Si1 ajSi1 u

aj1Si ajSi u

aj1 aj

Si1 Si > 0The inequality follows as, by assumption, Si1 > Si and aj1 > aj. Itfollows that we have positive assortative matching. h

Proof of Proposition 2. Bank i 1 will be willing to bid for thebanker of rank i a bonus qi1;i given by (11) asqi1;i 1 ai1=ai1 qi1. This is the marginal bid for bankeri. Hence bank i will match the marginal bidder:

aiqiSi aiqi1;iSi1 ai ai1 Si1 ai1qi1Si1 12It follows, by induction that aiSiqi

PNji1Sj aj1 aj

SNaNqN .The ultimate outside option of leaving the industry for all the bank-ers is normalised to 0 which yields qN 0. The result follows. h

Proof of Proposition 3. We rst show that a bank will pay alower bonus rate to the banker they hire than they would bidfor a better banker. This follows from (11) as qi;i1 qi 1 qi 1 ai=ai1 > 0. Hence a cap will be binding on a banksbidding for better staff.

Suppose that the cap affects the bidding of bank j for the betterbanker j 1 for the subset of banks j 2 M. If bank j 2 M then the bidfor banker j 1 is a bonus qj;j1 v as the cap is binding. Hencebank j 1 will secure banker j 1 at a bonus such that it matchesthe utility offered by bank j : aj1Sjv aj1Sj1qj1, yieldingqj1 v Sj=Sj1

< v 13expected utility would be, from (2):

Vij aj 1 qi;j

Si kSiG gaj 1 qi;j !c 9

Hence bank i is willing to bid up to a bonus of qi;j1 for banker j 1where:

Vij aj1 1 qi;j1

Si kSiG gaj1 1 qi;j1 !c 10Belgium, the CFA London, the Atlanta Fed, WBS University of War-wick, University of Zurich, and Christ Church, Oxford University.This work does not reect the view of the Bank of England orany other named individuals. Any errors remain my own.

Appendix A. Omitted Proofs

Proof of Lemma 1. Consider banks i and i 1 and bankers j andj 1. We wish to show that the bank with the larger pot of assetsin this business unit will secure the better banker. Suppose theoutside option of banker j is u. If bank i hires banker j at a bonus

J. Thanassoulis / Journal of BankIf instead a bank ranked jwere competing against a bank unaffectedby the cap, then the required bonus will also be unaffected by thecap, and is given by (12).

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004Suppose now the cap is so stringent that it affects more banks.Thus suppose the identity of the highest rank bank, with assetssmaller than i, which is affected by the cap becomes bank ~m wherei < ~m < m. The bonus payable by bank i can therefore be written,from (12) as

aiSiqi X~m1ji1

Sj aj1 aj a ~m1q ~m1S ~m1

The proof now follows by observing that the bonus bank ~m 1 paysdeclines as a result of the cap now affecting bank ~m. This follows asthe bid of bank ~m for banker ~m 1 is reduced by the cap. Hence thebonus paid by i again moves monotonically in v. This delivers theresult.

Finally, as the cap applies to all banks, the positive assortativematching result of Lemma 1 is unaffected. There is no re-ranking ofthe banks and so the allocation of bankers to banks isunaffected. h

Proof of Proposition 4. Given the maximisation problem (4) for-

mulate the Lagrangian L v b; xD E

g U x;qD E

; x;Vxh i

Rh i

with Lagrange multiplier g. The rst order condition then yieldsan expression for the optimal allocation x:

vb g @U@lq 2

@U@r2 Vx

0

Hence we have

x 12g @U=@r2 V

1 g@U@l

q vb

The direction of the vector x varies in the cap v unless b is propor-tional to q yielding the result. h

Proof of Proposition 5. First we note that both banks choosingexactly the same allocation in all asset classes so that they setS T=2 is not an equilibrium. As the banks are equal in size, com-petition for the a-banker would push their expected pay up to thepoint where both banks were indifferent between the a and bbanker. Thus it would be as if both hired b-bankers. This isXm1ji1

Sj aj1 aj am1vSm by 13 14

As the cap is binding on bank m by assumption, we have

v < quncappedm1 . Hence the bonus paid by bank i declines as a resultof the cap. The risk of a bank incurring a default event isG g=a 1 q c. As the bonus q declines this probability alsodeclines. The value of the bank rises by inspection of (2). Hencewe have the rst result.

We now turn to the second result. We wish to show that thebonus payable by bank i declines as the cap, v, falls. Suppose rstthat a reduction in the cap v does not alter the identify of thehighest rank bank, with assets in this business line smaller than i,which is affected by the cap. If so the bonus bank i pays is given byWe can now determine the equilibrium bonus paid by any banki. Let bank m be the bank with the greatest assets, conditional onbeing smaller than bank is, which is affected by the cap. Thusm 2 M and m > i. From (12) we have

aiSiqi Xm1

Sj aj1 aj am1qm1Sm1

& Finance xxx (2014) xxxxxx 11dominated by one bank moving e of their balance sheet to one ofthe business lines. They would then secure some benet from ana-banker which increases their prot.

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

The bank with the smaller asset allocation in any given classwill have T S in the asset class. If the cap on remuneration isbinding v < 1 b=a then the bonus is limited and so the bid iscapped at expected remuneration of av T S . Hence the bankwith the larger pot of assets secures the a-banker by offering a

c

a 1vTrSrc2 S2w

12 J. Thanassoulis / Journal of Bankingbonus rate of q v T S =S. The bank with a smaller pot of assetsin any given class will recruit the b-banker for a bonus of 0 as theoutside option is normalised to 0.

To identify the optimal asset allocation Swemust ensure there isno incentive tounilaterallydeviate to adifferent allocation eS. Denotethe value from such a deviation by V eS; T S where the secondargument captures the rivals weight in the asset class. From (5):

V eS; T S a 1 v T SeS

eS keSG g

a 1 v TSeS

0BB@1CCA

c

|{z}i

ceS T eS b T eS k T eS G gb

c15

We rst establish that the value function, (15) is concave in theasset allocation eS. This follows if the term i is convex in eS. To testthis dene h eS byh eS : g

a av TSeSThis is a hyperbola in eS. Consider the arm in which eS > v T S which is the relevant one as eS > T S. This curve is positive, down-wards sloping and convex. Now consider f eS eS h eS h ic. As cP 1a sufcient condition for this curve to be convex is if

0 < 2h0 eS eSh00 eS 2gv T S a ~S v T S 2 eS 2gv T S

a eS v T S 3() 0 < 2g v T S 2 which is true:

Thus the objective functionof the bank is concave and sohas a uniquemaximand given by the rst order condition. Hence an equilibrium isachieved when @V=@eS evaluated at eS S equals zero. This gives:@V

@eS S; T S 0 a c T 2S b kG gb c

kG ga 1 v TSS

!c 1 c v TSS

1 v TSS

( ) 16

This denes the equilibrium level of assets in the two businessunits, S and T S, implicitly as a function of v.

We wish to determine the change in the asset allocation to theover-weight asset in equilibrium. We have @V S v ; T S v =@eS 0 which denes S as a function of v. We therefore have0 @

2V

@eS@v dSdv @2V S; T S

@eS@eS @2V S; T S @eS@ T S

( )By algebraic manipulation of (16), @2V @eS@v. > 0.16 Due to theconcavity of the value function with respect to eS, we have that16 The result follows if ga 1v

c1 c v1vn o

is decreasing in v. Differentiating withrespect to v yields

g c v cv" #

c

a 1 v 1 v 1 v 2

And multiplying through by 1 v 2 conrms that the derivative is negative.

Please cite this article in press as: Thanassoulis, J. Bank pay caps, bank risk, a10.1016/j.jbankn.2014.04.004Sw

Simplifying, for c large enough we guarantee that dSw Sr =dSr > 1.In particular a sufcient condition for the result to hold is (6)0 a b c Tw 2Sw kG gb

kG ga

c 11 v TrSrSwh i

0@ 1Ac1 1 c 1 v Tr SrSw

18

I now show that the best response curve, Sw Sr is downwards slop-ing to yield the required result. Taking differentials we have

@

@S2wVw Sw; Tr Sr dSwdSr

@

@Sw@ Tr Sr Vw Sw; Tr Sr 0

Due to the concavity of the value function with respect to Sw,

@2V @S2w.

< 0. By algebraic manipulation one can conrm that

@2Vw @Sw@ Tr Sr = > 0 which implies that dSw Sr =dSr < 0 asrequired. h

Proof of Proposition 7. Lemma 6 shows that the asset allocationdecisions are strategic substitutes. We rst show that decreasingthe bonus cap v pushes the reaction function of bank w down.Taking differentials, @

@S2wVw dSwdv @

2

@Sw@vVw 0. By concavity@

@S2wVw < 0, and @

2

@Sw@vVw > 0 from the proof of Proposition 5 (using

Footnote 16). Hence dSw=dv > 0 as required. As the bonus cap doesnot apply to retail banking, the reaction function of bank r; Sr Sw isunaffected by v.

Reducing v will push the intersection of the reaction curvestowards greater retail banking assets if the equilibrium is stable(Tirole (1988)) so that 1 < dSi Sj

=dSi < 0 for all i j. This can be

conrmed by explicit differentiation of (18):

02cdSw Sr dSr

kG ga

cc c1 1

1vTrSrSwh ic2v2 TrSr 2S3w dSw Sr dSr

kG g c

c c1 1h i v2 TrSr@2V @eS@eS. < 0. By the same logic as for @2V=@eS@v we have@2V @eS@ T S . > 0. Combining we have determined thatdS=dv > 0, so the result is proved. h

Proof of Lemma 6. I will prove the result for bank w, the result forbank r follows analogously. Bank r is subject to a bonus cap of v inits bidding for the a-wholesale banker. This yields expected remu-neration of av Tr Sr . Hence bank w secures the a-wholesalebanker by offering a bonus rate of q v Tr Sr =Sw. Hence from(5) the value of bank w is:

Vw Sw;TrSr a 1vTrSrSw

SwkSwG g

a 1vTrSrSwh i

0@ 1Ac|{z}

i

b TwSw k TwSw G gb c

cSw TwSw 17

By the proof of Proposition 5 the value function Vw Sw; Tr Sr isconcave in the asset allocation Sw. Hence the best response of bankw is given by the rst order condition, @Vw=@Sw 0. Analogously to(16):

& Finance xxx (2014) xxxxxxusing the fact that Ti Si < Sj for ij by construction. The proofthat dSr Sw =dSw > 1 is analogous. Hence we have the desiredresult. h

nd macroprudential regulation. J. Bank Finance (2014), http://dx.doi.org/

Proof of Lemma 8. First we demonstrate that the bank wouldsecure the better wholesale banker. Suppose, for a contradiction,that the bank selects Tr Sr < Sh assets for its wholesale bankingbook. In this case the value of the bank is given by (8). This is con-cave in Sr . The rst order condition for this expression would setSr Syr . Hence bank r would have assets Tr Syr in its wholesalebanking book. But this is in excess of the hedge funds assets, Shby (7). Hence we have a contradiction and so the bank must prefer

0 1c

aqh;1Sh a b Sh. To hire the better executive the bank needs to

the hedge fund. The bank will therefore secure the b-banker to runits banking book. In this case the banks value is given by (8). Opti-mising this value over the asset allocation, the optimal wholesalebanking book size is then given as Syr . The wholesale banking bookhas shrunk and the banking book grown by comparison with thebound in Lemma 8. h

J. Thanassoulis / Journal of Banking & Finance xxx (2014) xxxxxx 13match this remuneration. This occurs if aqr Tr Sr P a b Sh.If the remuneration cap is binding on the bank then the betterexecutive can only be hired if Tr Sr P 1 b=a Sh=v . Supposethe cap is sufciently severe that the wholesale banking book isoptimally below this level.17 In this case the bank cannot outbid k Tr Sr G ga 1 1 b=a ShTrSr @ A

|{z}i

aSr

kSrG ga c

c Sr Tr Sr 19

This is concave in Sr if i is convex, which is true by the method ofproof of Proposition 5. Hence the objective function of the bank isconcave and so has a unique maximand given by the rst order con-dition. The rst order condition with respect to Sr delivers

ddSr

Vr Sr ; Sh kG ga 1 1 b=a ShTrSr

0@ 1Ac

1 c 11 1 b=a ShTrSr 1 b=a Sh

Tr Sr

24 35 kG g

a

c c Tr 2Sr

Algebraic manipulations deliver that ddSr Vr Syr ; Sh

< 0 using (7).

Hence the optimal size of the wholesale banking book is greaterthan Tr Syr , and so the assets devoted to retail are below Syr , yield-ing the result. h

Proof of Proposition 9. The hedge fund is willing to bid up to abonus given by (11) as qh;1 1 b=a. Hence the hedge fund wouldbe willing to offer the a-banker an expected utility of up toan asset allocation to wholesale banking which was sufcient tosecure the better banker.

With no remuneration caps the banker would be paid a bonusrate of 1 b=a Sh= Tr Sr , which follows from (12). Bank rsexpected value is then, adapting (17):

Vr Sr; Sh a 1 1 b=a ShTr Sr

Tr Sr 17 This is true in the limit as v tends to 0. Therefore there is a range of bonus caps forwhich it is true by continuity.

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Bank pay caps, bank risk, and macroprudential regulation1 Introduction2 Literature review3 The model3.1 Discussion of key assumptions

4 The no intervention benchmark5 Effect of a pay cap in proportion to (risk w