The impact of bank capital requirements on bank risk: an econometric puzzle and a proposed solution

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1 Banks and Bank Systems, Volume 4, Issue 1, 009 Robert L. Porter (USA) The mpact of bank captal requrements on bank rsk: an econometrc puzzle and a proposed soluton Abstract The relatonshp between bank rsk and bank captal has been frequently dscussed n the lterature on bankng. There s, however, an engma remanng. Emprcal studes have been nconsstent n ther measurement of the mpact of ncreased captal requrements on bank rsk. Do hgher mandatory captal requrements reduce rsk n bankng or do they actually ncrease bank rsk? The problem stems from the basc endogenous relatonshp between rsk and captal. In order to regress rsk on captal we need an nstrument for captal, but t s dffcult to fnd an nstrument that s related to a bank s captal that s not also related to the bank s rsk. In ths paper, we propose a soluton. Usng stochastc fronter analyss we develop an exogenous nstrument for captal that s closely correlated wth captal but not correlated wth rsk. We argue that ths nstrument wll be a useful contrbuton to the analyss of ths mportant topc. Keywords: bank captal adequacy, rsk management, captal requrements. JEL Classfcaton: G1, G8, G3. Introducton Rarely has the mportance of bank captal requrements receved more attenton than n the past 18 months or so. The sub-prme mortgage debacle produced bllons of dollars of losses that sgnfcantly reduced the captal of many large banks. Bank efforts to replace ther lost captal contnue today. Bank captal adequacy, however, s not a new topc. Banks have always receved specal attenton due to ther ablty to create money and due to the mpact that bank nformaton producton and lqudty servces have on the real economy. The prmary purpose of bank regulaton s to lmt the negatve externaltes arsng from bank falures and the prmary tool of choce by regulators s mnmum captal requrements. Complcatng the problem of bank solvency s the perverse nature of the federal bank safety net, especally depost nsurance. Snce depostors are not worred about bank rsk due to a government guarantee, banks are exempt from the normal dscplne credtors exercse over ther debtors. More mportantly, n the face of non-rsk based nsurance premums banks are drectly motvated to maxmze shareholder value by ncreasng rsk. Ths s a classc moral hazard problem. Agan, the tradtonal tool employed by bank regulators to deal wth ths problem has been the establshment of captal adequacy requrements. Arguments have been made, however, that mandatory captal requrements actually ncrease bank rsk. In other words, as captal requrements are ncreased, banks ncrease ther rsk n order to earn the same return on captal. Emprcal studes, however, of bank captal and bank rsk face an nherent problem. In order to measure the effect of the level of captal on bank Robert L. Porter, 009. rsk-takng t would be useful to regress rsk, as the dependent varable, on captal as the ndependent varable. However, there s an obvous endogenety problem. The amount of rsk you can undertake s dependent on the amount of captal you have and the amount of captal you need s dependent on the amount of rsk you want to undertake. In other words, they are jontly determned, much lke prce and quantty n a basc mcroeconomc analyss. The soluton to ths problem s normally to use ether a smultaneous equaton model or to use nstrumental varables. However, a smultaneous equaton model must be properly dentfed and no one has yet been able to accomplsh that n regard to rsk and bank captal. Lkewse no one, to our knowledge, has yet found a true nstrument for captal that s ndependent of rsk. We present a methodology for the development of an exogenous nstrument for captal n a regresson wth rsk. We propose to use stochastc fronter analyss to determne the maxmum possble ncome that can be acheved from a gven level of assets. Ths s referred to as fttng an upper envelope. Such a fronter s obvously exogenous to any specfc bank because t s determned by the data from all banks n the sample. The dstance from the fronter to any specfc bank actual ncome can be consdered a measure of bank neffcency. In other words, ths s a measure of how close the bank comes to maxmzng ts ncome based solely on the amount of assets employed. In order to develop an nstrument for captal we propose to create a second fronter condtoned on bank captal as well as the amount of assets employed. The ncremental neffcency from the second fronter s a functon of the bank s captal but ndependent of the bank s rsk, and t s ths ncremental neffcency that we propose to use as an nstrument for captal. 55

2 Banks and Bank Systems, Volume 4, Issue 1, Captal regulaton Brefly let us revew the current state of captal regulaton. In 1988 the Basle Commttee ssued a captal measurement system usually referred to as the Basle Captal Accord or Basle Accord I. Ths was n response to the deteroratng captal poston of many nternatonal banks at a tme of perceved ncrease n rsk. A key element of the Accord was a system of rsk weghtngs based on the type of assets held by a bank. In other words, two banks of the same sze would need dfferent amounts of captal based on the dfferent levels of rsk n ther assets. In addton, there was a unform defnton of what consttuted captal ; e.g. subordnated debt, loan loss reserves, etc. Fnally, off-balance-sheet tems were ncluded n the calculaton of requred captal as well as on-balance-sheet tems. After makng these calculatons, a bank was requred to have captal equal to 8% of the rsk weghted assets. The Accord appears to have been effectve because by 1993 bank captal ratos had ncreased to 8.01% from 6.1% n Modfcatons were made to the Basle Captal Accord durng the 1990s. A sgnfcant change ncorporated the problem of market rsk, n addton to credt rsk, nto the analyss of captal adequacy. Currently a new Captal Accord s scheduled for a phase startng n 008 and s generally referred to as Basle II. It represents an effort to fx some of the problems assocated wth the orgnal accord. One major ssue has been the potental for captal arbtrage between the orgnal rsk categores of Basle I. It s generally agreed that the orgnal rsk categores were too broad. Assets wth sgnfcantly dfferent rsk parameters nevertheless requred the same amount of captal to support them. For example, all commercal and ndustral loans requred the same amount of captal.. Lterature revew As far back as 1977, wrters have ponted out the ncentves banks have to ncrease rsk. Merton s 1977 artcle derves a formula to calculate the "far" premum for depost guarantees by the FDIC usng what are now famlar opton prcng technques. He fnds that the value of the premum s dentcal to a put opton where the value of the bank debt s the strke prce and the maturty s the next FDIC examnaton date. We know ncreasng rsk wll ncrease the value of optons but because depost nsurance premums were not based on rsk, banks could, n effect, transfer wealth from the FDIC to shareholders by ncreasng bank rsk. Merton s formula also allows us to measure the level of rsk n a bank and the drecton of any rsk change by calculatng the value of the depost nsurance. If the value 56 of the nsurance goes up the bank has become more rsky. If the value of the nsurance goes down, the bank has become less rsky. In 1984 Marcus and Shaked took Merton s theoretcal formula and made t operatonal for actual bank data. However, ther fndngs at the tme ndcated that f anythng, the then current depost nsurance premum was too hgh, not too low. In another study, Keeley (1990) rased the queston as to why t was not untl the 1980s that banks started explotng the federal safety net. Hs concluson s that a bank s charter value, when t s hgh, serves to mtgate rsk. In other words, banks lmt the amount of rsk they undertake n order to preserve ther charter value. Keeley then documents the change n bank charter value from a premum over book value to a dscount under book value. It was ths declne n charter value durng the 1980s that precptated the explotaton of depost nsurance. In a 1995 artcle, Berger, Herrng and Szego revew bank captal ratos from 1840 to In 1840 bank captal funded approxmately one-half of bank assets. In 1863, the frst Natonal Bankng Act created Natonal Banks and the offce of the Controller of the Currency to supervse these banks. Ths was perceved as reducng the nherent rsk n the new natonal banks and so captal ratos dd not need to be as hgh as they had been. In the creaton of the Federal Reserve System that ncluded the lender of last resort functon further reduced rsk and bankng and captal ratos declned agan. In 1933 the FDIC was created and Regulaton Q was promulgated to lmt the nterest rate a bank could pay on ts deposts. Captal rates declned agan. By 1989 the equty-to-asset rato of banks was generally only a lttle over 6%. It should be noted that n ths artcle the authors pont out that excessvely hgh captal requrements can produce socal costs through lower levels of ntermedaton. In addton, there can be unntended consequences such as rsk arbtrage (ncreasng rsk to offset the ncrease n captal and mantan the same return on captal), ncreased securtzaton, offbalance-sheet guarantees, etc., all of whch could mtgate the benefts of ncreased captal standards. It also needs to be noted that not everyone s n agreement wth the use of captal requrements to mtgate rsk n bankng. Berger et al., as dscussed above, ncluded comments on potentally bad unntended consequences. In addton, Km and Santomero (1988) argue that a smple captal rato cannot be effectve and any rato would need to have exactly correct rsk weghts n a rsk based system. Rochet (199) agrees. John, Saunders, and Senbet (000) argue that a regulatory emphass on captal

3 Banks and Bank Systems, Volume 4, Issue 1, 009 ratos may not be effectve n controllng rsk. Snce all banks wll have a dfferent nvestment opportunty set, an effcent allocaton of funds must ncorporate dfferent rsk takng for dfferent nvestment schedules. These authors go on to argue that senor bank management compensaton contracts may be a more promsng avenue to control rsk usng ncentve compatble contracts to acheve the optmal level of rsk. There are also other alternatves to mandatory captal requrements that could be used to lmt rsk n bankng. Prescott (1997) revews the pre-commtment approach to rsk management. Brefly, banks commt to a level of captal and f that level proves to be nsuffcent the bank s fned. Ths s used currently n the area of captal n support of a tradng portfolo but cannot be used for overall captal ratos snce a fne aganst a faled bank s not effectve. Gorton and Pennacch (199) dscuss narrow bankng. Ths proposes splttng the depost servces of banks from the credt servces. In other words, the fnancal system would nclude money market accounts and fnance companes. The money market accounts would only nvest n short-term hgh qualty assets and leave the lendng to the fnance companes that would not take n any deposts. Esty (1998) studes the mpact of contngent lablty of stockholders on rsk. In the late 19 th and early 0 th century bank stockholders were subject to a call or an assessment for more money f needed to meet the clams on a bank. There was a negatve relaton between ncreases n rsk and the possble call on bank stockholders. Calomrs (1999) makes a strong case for requrng the use of subordnated debt n bank captal structures. The need to ssue un-guaranteed debt and the assocated market dscplne would act as an effectve lmt to the amount of rsk a bank would be able to assume. 3. Emprcal studes Emprcal studes have been nconclusve n determnng the mpact of captal regulaton on rsk n bankng. Duan, Moreau and Sealey (199) address the queston of rsk-shftng to the FDIC drectly. Merton s 1977 artcle establshed that the partal dervatves from the opton prcng equaton wth respect to both varance and leverage are postve. The authors test ths wth real data and fnd that rsk-shftng behavor s very lmted. However, Hovakman and Kane (000) use the dentcal emprcal desgn and obtan opposte results. Interestngly, they also fnd evdence of a dchotomous strategy by banks. Hghly levered banks tend to have hgh rsk-shftng ncentves whle low leverage banks have low rsk-shftng ncentves. Ths supports Marcus (1984) who frst argued that banks are forced to choose between a hgh-rsk strategy and a low-rsk strategy because a mdrange polcy s sub-optmal. Hughes, Lang, Moon, and Pagano (003) also provde evdence of ths. In a measure of bank effcency, they document that hgh-leverage banks mprove ther effcency by ncreasng ther leverage further whle low-leverage banks mprove ther effcency by decreasng ther leverage. Emprcal studes that are drectly on pont nclude Ambrose, LaCour-Lttle and Sanders (003), Van Roy (003), and Alexander and Baptsta (006). In a study of securtzaton, Ambrose et al. present evdence that lenders retan hgher rsk loans n ther portfolo and sell lower rsk loans n the secondary market. Ths practce represents regulatory captal arbtrage. On the other hand, Van Roy (003) argues that hs evdence documents that the 1988 Basel Accord dd not result n banks takng on hgher levels of rsk. Fnally, Alexander and Baptsta (006) argue that a VaR constrant mposed on bank tradng portfolos can produce a perverse effect, namely that some banks may end up choosng rsker portfolos. It s apparent that ths topc contnues to generate nterest. It s our argument that the proposed methodology wll be a contrbuton to the lterature n ths feld. 4. Data and methodology We use a panel of data for bank holdng companes for our analyss. The perod covered s from 1993, the year after Basel I was fully mplemented, to 007. The number of banks n the sample ranges from 1,618 banks n 1993 to 964 banks n 007. The total of bank-year observatons s n excess of 5,000. The balance sheet data requred are taken from the Federal Reserve Bank Form 9-Y. Table 1 (see Appendx) dsplays summary statstcs on the data. To be effectve, an nstrument used n econometrc analyss needs to be hghly correlated wth the ndependent varable n queston, n ths case captal, but not correlated wth the dependent varable beng studed, n ths case rsk. We proceed as follows. Estmatng producton functons s standard fare n econometrcs and a fronter producton functon smply represents the maxmum output possble for a gven level of nputs. To estmate the fronter we must be consstent wth the proposton that all observatons fall below the fronter. Ths requres us to use a regresson that s ft to the data such that all observatons do ndeed fall below t. Formally, we use stochastc fronter analyss to develop a fronter, or upper envelope, of the pre-tax ncome earned based on the book-value of bank assets. 57

4 Banks and Bank Systems, Volume 4, Issue 1, 009 The estmate of a fronter s usually only a frst step n calculatng the effcency of a cross-secton of frms. The queston we are askng s: How effcent s a bank n convertng the assets wth whch t has to work nto pre-tax ncome? We argue that the fronter so developed s exogenous to any specfc bank snce t s based on the results of all banks n the sample. From the fronter we measure the neffcency of each bank as the dstance between the fronter and that specfc bank s pre-tax ncome. Ths measure, however, must be adjusted for those elements that are beyond the control of the bank,.e. the elements of luck. The specfcatons of our unrestrcted model are as follows: PTI 1BVA ( BVA) v u, v ~ d 58 v N(0, ) u( 0) ~ d u N(0, ) Where PTI = Pre-tax Income; BVA = Book Value of Assets; v = statstcal nose (luck); u = systematc shortfall (under management control). Please note that a quadratc specfcaton s used to allow for a non-lnear relaton between the pre-tax ncome and the book value of assets. The fronter value s the determnstc kernel of the stochastc fronter: FPTI BVA ) ( BVA ). ( The stochastc fronter value s the determnstc kernel plus the two-sded error: SFPTI FPTI v. Therefore, we can see that: u SFPTI PTI FPTI ( PTI v) and. E( u ) FPTI ( PTI E( v )) Snce the condtonal expectaton cannot be observed we must estmate t. The detals of fttng stochastc fronters can be found n Jondrow, Lovell, Materov, and Schmdt (198) and Greene (1997). Ths unrestrcted fronter model determnes the hghest potental pre-tax ncome based solely on the book-value of assets employed. Now we want to narrow the defnton and base the fronter value on the level of captal as well as the amount of assets. The mplcaton of usng the unrestrcted model s that we are measurng the uncondtonal neffcency of the bankng organzaton. By condtonng the model on captal we can develop a measure of the ncremental neffcency of an organzaton due to ts captal level. It s ths ncremental neffcency due to a bank s captal level that we propose to use as the nstrument for captal n a regresson of rsk on captal. Our restrcted model s specfed as follows: PTI 1 BVA ( BVA) 3BVC, where: PTI = Pre-tax Income; BVA = Book Value of Assets; BVC = Book Value of Captal; Epslon = as specfed n the base model. Agan, note that a quadratc specfcaton s used to allow for a non-lnear relaton between pre-tax ncome and the book value of assets. We save the estmate of u, the neffcency level of each bank, from both the unrestrcted model and the restrcted model. We then calculate the ncremental level of neffcency by subtractng the restrcted model results from the unrestrcted model results. Ths change n neffcency s due to the level of bank captal and s our nstrument. 5. Results We estmate our unrestrcted model. The results are shown n Table (see Appendx). As noted n our dscusson of the basc stochastc fronter model, the composte error term, epslon = v u, s asymmetrc and non-normal. The term v s a two-sded error term representng the varance from the fronter value due to factors over whch the bank has no control. Consstent wth accepted practce we assume a normal dstrbuton for v. The term u s one-sded and represents the short-fall n fronter value due to factors over whch the bank does have control. Ths s our measure of neffcency and, agan consstent wth common practce, we assume a half-normal dstrbuton for u. It s apparent that the asymmetry of epslon s due to u. The parameter, Lambda, s a measure of the asymmetry and s calculated by dvdng sgma u by sgma v. When Lambda equals zero, epslon = v and s normally dstrbuted. Whle there s no generally recognzed level of Lambda that can be referenced, the statstcal sgnfcance of Lambda provdes support for a stochastc fronter specfcaton. The results presented n Table show hghly sgnfcant parameters for both total asset varables. In addton, we fnd a hghly sgnfcant Lambda. We estmate a restrcted model usng the same data. Our results are shown n Table 3. The results agan show hghly sgnfcant parameters for both total asset varables. The coeffcent on captal s postve, a somewhat surprsng result, and hghly sgnfcant. We agan fnd a hghly sgnfcant Lambda. In order to complete the development of our nstrument for captal we save the estmated neffcences

5 from both models. We then take the dfference between the two estmated neffcences and employ t as our nstrument. To valdate the nstrument we regress our total captal varable on our nstrument and the other explanatory varable, total assets. We fnd a hghly sgnfcant coeffcent on our nstrument. We also note an ncrease n the F-value when we nclude the nstrument n the regresson compared wth the same regresson wthout the nstrument. The results are shown n Table 4. Concluson and further research The mportance of the relatonshp between bank rsk and bank captal has been well documented n the lterature. In ths paper we have developed an nstrument for captal to be ultmately used n a regresson of rsk on captal. Such an nstrument would be a soluton to the obvous endogenety problem assocated wth rsk and captal. We use stochastc fronter analyss to create an upper envelope of the pre-tax ncome generated from the book-value of that bank s assets and measure the level of neffcency for each bank n our sample. References Banks and Bank Systems, Volume 4, Issue 1, 009 We then create a second fronter condtoned on each bank s captal level as well as the book-value of ther assets and agan measure the level of neffcency of each bank. The dfference between these levels of neffcency s our nstrument. We argue that the ncremental neffcency s exogenous to the rsk of any specfc bank but correlated to that specfc bank s captal. We valdate the nstrument by measurng the sgnfcance of the nstrument n a regresson of the nstrument on the endogenous varable and the other explanatory varables. There are numerous further research topcs that can be pursued n lne wth ths analyss. The frst, and most obvous, s to employ the nstruments n a regresson of rsk on captal. Another approach would be to lmt the sample to publcly owned bank holdng companes and use the market-value of assets as the dependent varable n leu of pre-tax ncome. Ths would produce a fronter based on the maxmum market-value of assets that can be attaned wth a gven level of book-value assets. Fnally, f the stochastc fronter analyss can be confdently used to develop nstrumental varables the applcaton of the process s extremely wdespread. 1. Alexander, Gordon J. and Alexandre Baptsta (006), Does the Basle Captal Accord Reduce Bank Fraglty? An Assessment of the Value-at-Rsk Approach, Journal of Monetary Economcs, Vol. 53: Ambrose, Brent W., Mchael LaCour-Lttle, and Anthony B. Sanders (003), Does Regulatory Captal Arbtrage or Asymmetrc Informaton Drve Securtzaton?, Unpublshed paper, Unversty of Kentucky and Oho State Unversty. 3. Berger, Allen N., Rchard J. Herrng and Gorgo P. Szego (1995), The Role of Captal n Fnancal Insttutons, Journal of Bankng & Fnance, Vol. 19: Calomrs, Charles W. (1999), Buldng an Incentve-compatble Safety Net, Journal of Bankng & Fnance, Vol. 3: Duan, Jn-Chuan, Arthur F. Moreau and C.W. Sealey (199), Fxed-rate Depost Insurance and Rsk-shftng Behavor at Commercal Banks, Journal of Bankng and Fnance, Vol. 16: Esty, Benjamn C. (1998), The Impact of Contngent Lablty on Commercal Bank Rsk Takng, Journal of Fnancal Economcs, Vol. 47: Gorton, Gary and George Pennacch (199), Fnancal Innovaton and the Provson of Lqudty Servces, Pp n Reform of Federal Depost Insurance, eds J. Barth and D. Brumbaugh. Harper Collns: New York. 8. Hovakman, Armen and Edward J. Kane (000), Effectveness of Captal Regulaton at U.S. Commercal Banks, 1985 to 1994, Journal of Fnance, Vol. 55: Hughes, Joseph P., Wllam W. Lang, Choon-Geol Moon and Mchael S. Pagano (003), Manageral Incentves and the Effcency of Captal Structure n U.S. Commercal Bankng, Unpublshed paper, Federal Reserve Bank of Phladelpha. 10. John, Kose, Anthony Saunders and Lemma Senbet (000), A Theory of Bank Regulaton and Management Compensaton, Revew of Fnancal Studes, Vol. 13: Keeley, Mchael C. (1990), Depost Insurance, Rsk, and Market Power n Bankng, Amercan Economc Revew, Vol. 80: Keeley, Mchael C. (199), Bank Captal Regulaton n the 1980's: Effectve or Ineffectve?, Pp n Bank Management & Regulaton: A Book of Readngs, eds A. Saunders, G. Udell and L. Whte. Mayfeld Publshng Company: Mountan Vew, Calforna. 13. Km, Daesk and Anthony M. Santomero (1988), Rsk n Bankng and Captal Regulaton, Journal of Fnance, Vol. 4: Marcus, Alan J. (1984), Deregulaton and Bank Fnancal Polcy, Journal of Bankng and Fnance, Vol. 8: Marcus, Alan J. and Israel Shaked (1984), The Valuaton of FDIC Depost Insurance Usng Opton-Prcng Estmates, Journal of Money, Credt and Bankng, Vol. 16:

6 Banks and Bank Systems, Volume 4, Issue 1, Merton, Robert C. (1977), An Analytc Dervaton of the Cost of Depost Insurance and Loan Guarantees, Journal of Bankng and Fnance, Vol. 1: Prescott, Edward S. (1997), The Pre-Commtment Approach n a Model of Regulatory Bankng Captal, FRB of Rchmond Economc Quarterly, Vol. 83: Rochet, Jean-Charles (199), Captal Requrements and the Behavour of Commercal Banks, European Economc Revew, Vol. 36: Van Roy, Patrck (003), The Impact of the 1988 Basel Accord on Banks Captal Ratos and Credt Rsk-takng: an Internatonal Study, Unpublshed paper, European Center for Advanced Research n Economcs and Statstcs. Appendx Table 1. Summary statstcs Followng s the summary of statstcs of key varables ncluded n the analyss. There are 5,066 observatons for each varable. Amounts are n mllons of dollars. Varable Mnmum Maxmum Total assets 7,850,187,631,000 5,650,096 50,088,304 Total equty -654, ,803,41 464,63 3,987,45 Pre-tax ncome -7,19,73 3,007,503 93, ,34 Unrestrcted neffcency 87,871 3,1, , ,860 Restrcted neffcency 61,108,40,535 70,189 95,985 Instrument for captal -1,631,83,83,134 95,444 8,580 Table. Unrestrcted fronter analyss The dependent varable s Pre-tax ncome. PTI BVA ( BVA) v u v ~ d N(0, ) u( 0) ~ d N(0, ), 1 v u where PTI = Pre-tax Income; BVA = Book Value of Assets; v = statstcal nose (luck); u = systematc shortfall (under management control). The model s estmated usng stochastc fronter analyss wth maxmum lkelhood estmates. The error term equals v mnus u. There are 5,066 observatons. Values n parentheses are t-values. Intercept Total assets Total assets squared Prmary varables Coeffcent 1,5,910 (111.19) (50.69) -4,316,000,000 (-68.46) Parameters for compound error Lambda (47.54) Sgma Table 3. Restrcted fronter analyss The dependent varable s Pre-tax Income. PTI BVA ( BVA) BVC v u v ~ d N (0, ) u( 0) ~ d N (0, ), 1 3 v u where PTI = Pre-tax Income; BVA = Book Value of Assets; BVC = Book Value of Captal; v = statstcal nose (luck); u = systematc shortfall (under management control). The model s estmated usng stochastc fronter analyss wth maxmum lkelhood estmates. The error term equals v mnus u. There are 5,066 observatons. Values n parentheses are t-values (69.5) Intercept Total assets Total assets squared Prmary varables Coeffcent 850,976 (113.0) (-1.808) -14,310,300,000 (-44.15) 60

7 Total captal Table 3 (cont.). Restrcted fronter analyss Banks and Bank Systems, Volume 4, Issue 1, ( Parameters for compound error Lambda (59.143) Sgma 907,01 ( ) Table 4. Valdaton of nstrument Dependent varable s Total Captal. Model 1 s the regresson of Total Captal on Total Assets. Model s the regresson of Total Captal on Total Assets and the Instrumental Varable calculated. Values n parentheses are t-values. Intercept Total assets Instrumental varable Varable Model 1 Model 7,466,455,916 (4.54) (156.09) (644.) (98.8) ( ) F-value 415,0 433,111 61