WORKING PAPER NO SAFETY IN NUMBERS? GEOGRAPHIC DIVERSIFICATION AND BANK INSOLVENCY RISK. Joseph P. Hughes Rutgers University

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1 WORKING PAPER NO SAFETY IN NUMBERS? GEOGRAPHIC DIVERSIFICATION AND BANK INSOLVENCY RISK Joseph P. Hughes Rutgers Unversty Wllam Lang Offce of the Comptroller of the Currency Loretta J. Mester Federal Reserve Bank of Phladelpha and The Wharton School, Unversty of Pennsylvana Choon-Geol Moon Hanyang Unversty May 1996 The authors would lke to thank Wllam C. Hunter, whose comments on a prevous paper nspred ths nvestgaton. The vews expressed n ths paper do not necessarly represent those of the Federal Reserve Bank of Phladelpha or of the Federal Reserve System or of the Offce of the Comptroller of the Currency.

2 SAFETY IN NUMBERS? GEOGRAPHIC DIVERSIFICATION AND BANK INSOLVENCY RISK Abstract The Regle-Neal Interstate Bankng and Branchng Effcency Act, passed n September 1994 and effectve June 1, 1997, wll allow natonally chartered banks to branch across state lnes. Ths act wll remove mpedments to nterstate expanson and permt the consoldaton of exstng nterstate networks. What wll be the mpact of ths legslaton on bank performance and bank safety? Removng mpedments to geographc expanson should mprove the rsk-return tradeoff faced by most banks. However, ths paper argues that economc theory does not tell us whether an mprovement n the rskreturn tradeoff wll lead to a reducton n the volatlty of bank returns or n the probablty of nsolvency. We nvestgate the role of geographc dversfcaton on bank performance and safety usng bank holdng company data. We fnd that an ncrease n the number of branches lowers nsolvency rsk and ncreases effcency for neffcent bank holdng companes; an ncrease n the number of states n whch a bank holdng company operates ncreases nsolvency rsk but has an nsgnfcant effect on effcency. Branch expanson rases the rsk of nsolvency for effcent bank holdng companes, whle an ncrease n the number of states has an nsgnfcant effect on nsolvency rsk.

3 SAFETY IN NUMBERS? GEOGRAPHIC DIVERSIFICATION AND BANK INSOLVENCY RISK Introducton In spte of consderable legal obstacles, banks n the Unted States have strved to cross state lnes. Despte the apparent desre by banks to expand geographcally, many studes of nterstate bankng have not found an nverse relatonshp between the volatlty of bank returns and measures of geographc 1 dversty. At frst glance ths mght seem to contradct the smple ntuton that geographc expanson wll lead to a decrease n the rskness of bank returns. Ths paper argues that an ncrease n the rskness of bank returns as a result of geographcal dversfcaton s consstent wth economc theory and s consstent wth the noton that geographcal dversfcaton can mprove bank effcency. Even f the volatlty of bank returns rses as a result of geographc dversfcaton, the probablty of bank nsolvency would fall f there were a suffcently large ncrease n expected returns. However, economc theory alone cannot tell us whether there wll be a rse or a fall n bank safety n response to mproved opportuntes to dversfy. 2 Ths theoretcal ndetermnacy emphaszes the need for emprcally estmatng bank responses to geographcal dversfcaton. Ths paper extends the emprcal work n Hughes, Lang, Mester, and Moon (hereafter referred to as HLMM) (forthcomng) by studyng the mpact of geographcal dversfcaton on bank nsolvency rsk on a sample of 443 U.S. bank holdng companes (BHCs) operatng n As n that paper, we employ a model of producton developed by HLMM (1995) that allows managers to trade return for reduced rsk. The model accommodates non-neutral preferences toward rsk whle allowng for rsk neutralty as a specal case. Ths provdes a test of the usual assumpton of neutralty. Based on the proft functon, the producton system employs the Almost Ideal Demand System to obtan the functonal forms for the proft and nput share equatons. In the case where managers are rsk neutral or, equvalently, proft s maxmzed, the functonal forms are dentcal to the standard translog proft system. Usng procedures developed by Hughes and Moon (1995), the estmated proft functon,

4 2 condtoned on equty captal, s used to obtan an expected rate of return on equty for each BHC n the sample. The standard error of the predcton proxes rsk and a stochastc rsk-return fronter s estmated to obtan a best-practce fronter. Varous measures of effcency, representng the dstance of each bank 3 from the best-practce, rsk-return fronter, are then computed. Fnally, the effects of geographc dversfcaton on expected return, rsk, effcency, and the probablty of nsolvency are estmated. The estmated effects of geographc dversfcaton on return, rsk, and safety depend on whether the BHC s neffcent or effcent. For neffcent BHCs, an ncrease n the number of branches n the BHC has a sgnfcant negatve effect on rsk wthout a statstcally sgnfcant effect on expected return and a sgnfcant postve effect on the effcency of neffcent BHCs. An ncrease n the number of states n whch a BHC operates has a sgnfcant postve effect on rsk, an nsgnfcant effect on expected return, and an nsgnfcant postve effect on effcency. A proportonate ncrease n number of states and branches has a sgnfcant postve effect on expected return, a sgnfcant postve effect on rsk (when BHC asset sze s allowed to vary along wth the number of states and branches), and a sgnfcant postve effect on effcency. For effcent BHCs, only the number of branches has a sgnfcant effect on expected return or rsk. An ncrease n the number of branches n the BHC has a sgnfcant negatve effect on expected return and rsk, movng the BHC downward along the effcent rsk-return fronter. The number of states, asset sze, and a proportonate ncrease n number of states and branches are nsgnfcant. Thus, n contrast to neffcent BHCs, geographc expansveness does not appear to be an advantage for effcent BHCs. For both effcent and neffcent BHCs, an ncrease n number of branches s sgnfcantly related to a decrease n bank safety. The effect of an ncrease n the number of states s of opposte sgn and s sgnfcant for neffcent BHCs. A proportonate ncrease n the number of branches and states, holdng BHC asset sze constant, has an nsgnfcant effect on nsolvency rsk. When BHC sze s allowed to vary, a proportonate ncrease n branches and states has a sgnfcant postve effect on nsolvency rsk for neffcent BHCs, reflectng the effect of an ncrease n BHC sze.

5 3 I. Geographcal Dversfcaton and Manageral Rsk Preferences The Regle-Neal Interstate Bankng and Branchng Effcency Act, passed n September 1994 and effectve June 1, 1997, opens new avenues to nterstate expanson and permts the consoldaton of exstng nterstate networks. Reduced barrers to geographc expanson are expected to enhance opportuntes for dversfcaton and reduce the operatng costs of exstng nterstate structures. Smple logc would seem to ndcate that such changes wll ncrease effcency and create a safer bankng system. Enhancng the dversfcaton of assets and labltes allows banks to mprove ther rsk-return tradeoff. In other words, at any gven level of rsk, return s mproved or, equvalently, at any gven rate of return, rsk s reduced. However, whle dversfcaton lowers the rsk of obtanng the bank s current level of returns, dversfcaton also changes the margnal compensaton for rsk-takng. At the margn, the extra returns that can be obtaned for an ncrease n rsk, the margnal compensaton for rsk, wll ncrease at any level of expected return. Equvalently, the prce of rsk s reduced. How do rsk-averse banks respond to a change n the margnal compensaton for rsk-takng? Does the dversfyng bank's demand for rsk ncrease or decrease? Ths depends on the magntude of the substtuton and wealth effects from a lowerng of the prce of rsk. The substtuton effect mples that the bank takes on more rsk n response to the lower prce. The drecton of the wealth effect can be postve or negatve dependng on the underlyng rsk preferences of the bank. Thus, the sum of the two effects depends on the bank's preferences for rsk and return. A rsk-averse bank that dversfes reduces ts prce of rsk and, n response, may ncrease ts rsk-takng or reduce t. Hence, the bank that dversfes doesn t necessarly lower the volatlty of ts returns. The effects on the rsk-return fronter of an ncrease n geographc dversfcaton should depend on whether banks are effcent or neffcent. Effcent BHCs operate on the envelope of the rsk-return fronters of all ndvdual BHCs (what we call the envelope fronter ), whle neffcent BHCs operate on fronters nsde the envelope (see Hughes and Moon, 1995).

6 4 An neffcent BHC that mproves ts dversfcaton should be able to reduce rsk at any gven level of return or, equvalently, to ncrease return at any gven level of rsk, except at the rsk-free return on government securtes. In other words, enhanced dversfcaton shfts the rsk-return fronter upward at all postve levels of rsk. As llustrated n Fgure 1, ncreased geographc dversfcaton rotates the fronter upward from ts ntersecton wth the vertcal axs (the rsk-free return), shftng the fronter upward and changng ts slope at any gven level of rsk. 4 How do neffcent BHCs that dversfy respond to the lower prce of rsk? One possble outcome s a movement from the orgnal pont A to pont D. Ths case generates the ntuton that mproved dversfcaton wll lead to lower rsk and hgher return. However, a new tangency at pont DN cannot be ruled out. The same analyss holds for a rsk-neutral bank, whch maxmzes return. At the maxmum return, the prce of rsk s zero, snce the ndfference curves are horzontal. Although dversfcaton necessarly ncreases the maxmum return, t s not clear, a pror, how t affects the level of rsk where return s maxmzed. Ths can be seen n Fgure 1 where the new maxmum s at a hgher level of rsk. What, then, consttutes evdence that greater geographc extensveness leads to greater dversfcaton for neffcent banks? If dversfcaton does not necessarly reduce rsk, t must ncrease return and mprove the effcency of neffcent BHCs. The mpact of geographcal dversfcaton on effcent BHCs dffers from the neffcent case. The BHCs operatng on the envelope fronter have effcently exploted ther chosen strateges of dversfcaton. Hence, a varaton n one of those strateges, such as geographc expanson, may have no effect on expected return and rsk or t may move the BHC along the envelope fronter. In the latter case, one would expect the geographcally dverse BHCs to be located n a specfc regon of the envelope, whle, n the former case, they should be scattered over the envelope. 5 Whle bank regulaton ams at mprovng bank effcency and proftablty, bank regulators are partcularly concerned wth the mpact of polcy changes on bank safety. Snce rsk measured relatve to

7 5 return s a component of a bank s rsk of nsolvency, enhanced dversfcaton also mples that the rsk of nsolvency s reduced at any gven level of return. Moreover, even f the volatlty of returns rses as a result of mproved opportuntes for dversfcaton, the rsk of nsolvency could stll fall f the postve mpact on bank returns s suffcently large. In other words, the condtons under whch nsolvency rsk would rse n response to ncreased dversfcaton are stronger than the condtons that lead to an ncrease n the rskness of bank returns. Theoretcally t s stll feasble for a bank to make choces that ncrease nsolvency rsk n response to mproved opportuntes to dversfy. One measure of the rsk of nsolvency s the nverse z-score,.e., the standard devaton of proft dvded by the sum of bank captal and expected proft, whch equals the standard devaton of return on equty dvded by (1 + expected return on equty). We use ths measure of (nverse) bank safety to show the effect of mproved dversfcaton on the safety of neffcent BHCs (Fgure 2), where return s measured as return on equty and rsk s measured as the standard devaton of return on equty. The so-nverse z-score contours, whch show all rsk-return combnatons that have the same nverse z-score, are lnear, and contours to the rght mply hgher nverse z-scores (hgher nsolvency 6 rsk). An neffcent BHC operatng at pont A that dversfes geographcally s able to choose from a range of more effcent producton plans on the hgher fronter. Ponts on the new fronter to the left of pont B are safer than the one at pont A. The BHC could, of course, choose a pont to the rght of B. Hence, there s no a pror predcton. The effect on safety of a change n the dversfcaton strategy of an effcent BHC depends on how the change shfts the BHC s poston on the envelope. From Fgure 2, t s clear that a movement along the fronter toward hgher return and hgher rsk must necessarly ncrease the nverse z-score and, thus, reduce safety. 7 II. Modelng Rsky Producton and Manageral Rsk Preferences We characterze bankng technology n terms of fnancal ntermedaton. Banks employ labor,

8 6 physcal captal, and varous fundng sources, ncludng equty (fnancal) captal, to produce nvestments n fnancal outputs, prncpally loans and securtes. Equty captal s not only a source of loanable funds, but t also serves as a cushon aganst loan losses that threaten the bank s solvency. Denotng the output vector by y, the nput vector by x, and equty captal by k, the bank s producton technology s MT MT represented by the transformaton functon T(y,x,k) = 0, where >0 and <0. My Mx j Any producton plan (y,x,k) nvolves a varety of rsks. The qualty of the assets and the ntensty of resources devoted to credt analyss and loan montorng shape the credt rsk. The duratons of assets and labltes determne the nterest rate rsk. The duraton of the labltes affects the lqudty rsk. The level of fnancal captal nfluences the rsk of nsolvency. Controllng rsks s costly. Addtonal resources devoted to credt analyss and loan montorng reduce credt rsk and ncrease return. The rsk-neutral bank nvests n these addtonal resources untl the net return s maxmzed. The rsk-averse bank s wllng to nvest even more ntensvely n these nputs to further reduce credt rsk. In dong so, though, t sacrfces return for reduced rsk. By the standard of rsk neutralty, t has "overemployed" these nputs. Smlarly, the rsk-averse bank mght employ more expensve but less volatle fundng sources to reduce lqudty rsk. It mght hold a hgher proporton of ts assets n securtes to reduce overall credt rsk, even at the expense of reduced profts. Thus, the bank managers most preferred producton plan does not necessarly maxmze proft. II.A. Manageral Utlty The nfluence of the bank managers preferences for rsk on the organzaton of producton can be represented by a manageral utlty functon that ncludes proft,, and the producton plan, (y,x,k). The ncluson of the producton plan n the utlty functon accounts for rsk and potentally non-neutral rsk preferences. Managers who are rsk-neutral maxmze expected proft. Managers who are rsk-averse accept lower proft for reduced rsk. Hence, the managers most preferred producton plan maxmzes

9 7 utlty but not necessarly proft. Hughes and Moon (1995) argue that the ncluson of the producton plan n the utlty functon, rather than expected proft and rsk, provdes a more general representaton of rsk and rsk preferences. Under certan condtons the more general representaton s equvalent to utlty expressed n terms of expected proft and rsk. Stochastc market condtons nteract n complex ways wth the feasble producton plans to determne the rsk nherent n producton. Gven these stochastc market condtons, any technologcally feasble producton plan entals an expected return and rsk. The realzaton of stochastc market condtons, represented by the state varable s, and the producton plan, (y,x,k), mply a realzaton of proft = g(y,x,k,s). Employng the probablty dstrbuton of s, a condtonal probablty dstrbuton for realzed proft, f( ;y,x,k), can be obtaned. 4 Hence, the producton plan (y,x,k) mples an expected proft, E( ;y,x,k) ' [f( ;y,x,k) ]d m and a level of rsk measured by the standard devaton S( ;y,x,k). The manageral utlty functon defned over proft and the producton plan ranks dstrbutons of proft. If managers are rsk-neutral, they rank plans only by the frst moment of the dstrbuton. However, f they are not rsk-neutral, hgher moments of the dstrbuton wll matter n the rankng. For certan probablty dstrbutons and utlty functons, utlty can be characterzed solely n terms of the frst and second moments. Thus, utlty defned over proft and the producton plan subsumes these cases and allows greater generalty. To account for the mportant detals of bank producton, other arguments are also ncluded n the utlty functon: U(,y,x,k,p,r,n), where p denotes the prces or nterest rates earned by the outputs, r s the rsk-free rate of return, and n s nonperformng assets. The prces of assets, relatve to the rsk-free rate of return, ndcate the rsk premum on assets, an ex ante measure of asset qualty. The level of nonperformng loans s a gauge of ex post asset qualty. &4 II.B. The Most Preferred Producton Plan

10 8 The managers most preferred producton plan s the one out of all feasble plans that maxmzes manageral utlty: max U(,x;y,p,r,n,k),x s.t. y % m & x & p ' 0 T(x;y,k) # 0 (1) (2) (3) where m denotes nonnterest ncome, p s the prce of a real dollar of after-tax accountng proft,, n terms of nomnal, before-tax dollars, p = 1/(1-t) where t s the margnal tax rate on proft and a real dollar s assumed equal to one nomnal dollar. Hence, p s nomnal, before-tax accountng proft, whch dffers from nomnal, before-tax economc proft, p, by the requred return on equty captal, wk, k p ' p % w k k ' p@y % m & w@x. (4) The problem of maxmzng manageral utlty n (1)-(3) s condtoned on the level of equty captal so that maxmzng utlty wth respect to accountng proft,, s equvalent to maxmzng t wth respect to economc proft and wth respect to proft dvded by equty, the rate of return on equty. It s condtoned on the output vector so that the most preferred cost functon can be computed (see HLMM, 1995 and forthcomng). For the purposes of dervng the expected return and rsk of the producton plan, the cost functon can be gnored. Other studes have also used captal as a condtonng argument. Hancock (1985, 1986) condtoned the proft functon on captal whle Hughes and Mester (1993) and McAllster and McManus (1993) constructed cost functons that controlled for captalzaton. Denotng the prce vector by v = (w,p,r,p ), the most preferred producton plan s defned by the utlty-maxmzng soluton x(y,n,v,m,k) and (y,n,v,m,k). The proft functon wll account for any tradeoff of proft for reduced rsk. Hence, t s suffcently general to account for rsk-averse as well as rsk-neutral preferences. Dvdng the proft functon by equty captal transforms t nto the rate of return on equty.

11 9 III. Dervng the Most Preferred Proft System from the Almost Ideal Demand System Characterzng the bank's management as maxmzng utlty subject to a technology constrant suggests that a functonal form for the problem mght be found n consumer theory. HLMM (1995) employ the Almost Ideal Demand System to obtan the most preferred proft system and show that n the case of rsk-neutralty, t s dentcally equal to the standard translog proft functon and share equatons. The Almost Ideal Demand System postulates a flexble functonal form for the consumer's expendture functon and employs Shephard's lemma to obtan the expendture-mnmzng demand functons. Substtutng the ndrect utlty functon, obtaned by nvertng the expendture functon, nto these demand functons, they are transformed nto utlty-maxmzng demands, expressed as shares n expendture. III.A. Adaptng the Almost Ideal Demand System Followng ths procedure, HLMM (1995) adapt the Almost Ideal (AI) expendture functon to represent the generalzed manageral preferences: ln E(@) ' ln P % U@ 0 y j w j j p µ k, (5) where ln P ' 0 % p ln p % j ln y % j j ln w j % ln p % ln r % hln n % ln k % 1 2 pp (ln p)2 % 1 2 j j j j ln y ln y j % 1 2 j s j t ( j ln w s ln w t % 1 2 (ln p ) 2 % 1 2 rr(ln r)2 % 1 h 2 nn (ln n)2 % 1 2 kk (ln k)2 % jj pj ln p ln y j % js ps ln p ln w s % p ln p ln p % pr ln p ln r % pn ln p ln n % pk ln p ln k

12 % jj j s js ln y j ln w s % j j j ln y j ln p % jj jr ln y j ln r % jj jn ln y j ln n % j jk ln 10 y j ln k j % 1 2 j s ( s ln w s ln p % 1 2 j s ( s ln p ln w s % js sr ln w s ln r % j s sn ln w s ln n % j s sk ln w s ln k % r ln p ln r % n ln p ln n % kln p ln k % rn ln r ln n % rk ln r ln k % h nk ln n ln k. To reduce the number of parameters to be estmated, the ndvdual output prces, p, are replaced y by ther weghted average, p' j p. j j y j Invertng the expendture functon yelds the ndrect manageral utlty functon, V(@) ' 0 ln(p@y % m) & ln P y j w j p µ k (6) Applyng Shephard's lemma to the expendture functon and substtutng the ndrect manageral utlty functon nto t yelds the utlty-maxmzng proft and nput demand equatons, expressed as shares n expendture, p@y + m, on proft, p, and the nputs, w@x: Mln E ' Mln w w x p@y % m ' Mln P Mln w % [ln(p@y % m) & ln P] ' % js s ln w s % p ln p % j j j ln y j % ln p (7) % r ln r % n ln n % k ln k % [ln(p@y % m) & ln P] Mln E Mln p ' p p@y % m ' Mln P Mln p % µ[ln(p@y % m) & ln P] ' % ln p % p ln p % jj j ln y j % js s ln w s (8) % r ln r % n ln n % k ln k % µ[ln(p@y % m) & ln P].

13 11 III.B. Dervng the Demand for Equty Captal Snce the problem of maxmzng constraned manageral utlty was condtoned on the bank's captalzaton, the most preferred level of equty captal can be obtaned by addng a second stage to the optmzaton. Hence, the condtonal ndrect utlty functon can be obtaned by evaluatng the Lagrangean functon for the utlty maxmzaton problem at the optmum, V(y,n,v,m,k) / U( (@),x(@);y,p,r,n,k) % (@)[p@y % m & w@x(@) & p (@)] (9) % (@)[T(x(@);y,k)]. and by maxmzng (9) wth respect to equty captal k, the demand for equty captal follows from the resultng frst-order condton, MV(@) Mk ' MV(@) Mln k Mln k Mk '& k 0 y 1 j j j p µ k Mln P Mln k % [ln(p@y % m) & ln P] ' 0 Y % kk ln k % pk ln p % jj jk ln y j % js sk ln w s % k ln p % rk ln r % h nk ln n % [ln(p@y % m) & ln P] ' 0. (10) III.C. The Case of Rsk Neutralty The standard translog proft (cost) functon s embedded n the most preferred proft system. When the rsk terms are omtted from the expresson, ln P becomes the standard translog cost functon, condtoned on equty captal. Hence, the frst term on the rght-hand sde of (8) resembles the translog

14 12 nput share equaton n the case of rsk neutralty. The second term s a normalzed revenue effect. The rght-hand sde of (9) contans the full expresson, ln P. These features of the most preferred proft system suggest that there are a set of parameter values mpled by rsk neutralty that reduce the most preferred system to the standard translog proft (cost) functon. If banks maxmze proft or, equvalently, snce the system s condtoned on equty captal, f they maxmze the rate of return on equty, varatons n the tax rate p wll have no effect on the choce of before-tax proft. Addtonally, the revenue and rsk characterstcs of producton represented by the output prce vector wll not nfluence the bank s cost-mnmzng producton plan (condtoned on the output vector). Fnally, a varaton n m wll have no margnal sgnfcance for the optmal nput demands. Thus, proft maxmzaton behavor mples a set of testable parameter restrctons derved n HLMM (1995). III.D. The Emprcal Model Nonlnear two-stage least squares are employed to estmate the system of equatons consstng of the proft share equaton (8), the nput share equatons (7), and the frst-order condton for captalzaton (10). Addng up condtons and certan symmetry condtons are mposed on the estmaton. Ther detals are dscussed n HLMM (1995 and forthcomng). IV. Dervng Rsk, Return, and the Best-Practce Fronter Predcted proft, p ^, from the estmated proft share equaton (8), condtoned on the level of equty captal, s dvded by captal to obtan an expected rate of return on equty, ER, for each BHC n the sample. The BHC s rsk, RK, s measured by the standard error of the predcted proft, dvded by captal, k. Both ER and RK are functons of (y,n,w,p,r,p,m,k). A best-practce rsk-return fronter s computed by regressng expected return rescaled by ts standard devaton, ERN, on a constant term, rsk rescaled by ts standard devaton, RKN, and on squared

15 13 rescaled rsk: ER ) ' 0 % 1 RK ) % 2 RK ) 2 %. (11) A composte error term, = + u, s employed to dstngush neffcency from statstcal nose. The two-sded component,, s dstrbuted N(0, 2 v ) and accounts for any unmeasured randomness n the data generaton process of rsk and return. The one-sded component, u < 0 s dstrbuted half normally,.e., t s the absolute value of a varable dstrbuted N(0, 2 u), and gauges neffcency, the falure to acheve the fronter return at a gven level of rsk. The log-lkelhood functon of ths fronter s ln L ' N 2 ln 2 & Nln & j '1 N 2 % j N '1 ln & (12) where N s the number of BHCs, 2 ' 2 u % 2 v, ' u, and (@) s the standard normal cumulatve dstrbuton functon. We estmate ths fronter usng maxmum-lkelhood technques. v V. Measurng Effcency BHCs that choose the producton plan that maxmzes return are, n the rsk-neutral sense, allocatvely effcent. BHCs that trade return for reduced rsk are, by contrast, allocatvely neffcent to the extent that they employ more expensve but less volatle fundng sources to reduce lqudty rsk and techncally neffcent to the extent that they devote extra labor to analyze credt applcatons and montor loans to reduce credt rsk. To the extent that they choose a less rsky, but less proftable mx of assets, they are allocatvely neffcent. The rsk-neutral concepts of allocatve and techncal effcency do not allow for the possblty that the reduced rsk may be acheved effcently, that s, at the maxmum return feasble at the gven level of rsk. Although the falure to acheve allocatve effcency may represent agency problems, t may also reflect regulatory ncentves. Snce rsk-takng by ndvdual banks generates externaltes affectng the payments system, regulators may nduce rsk-averse behavor n bank managers who would otherwse be

16 14 neutral toward rsk. Hence, when managers trade return for reduced rsk, t s mportant to account for ths tradeoff n measurng effcency. The rsk-return fronter shown n Fgure 3 llustrates effcency when managers choose producton plans that do not maxmze return. The allocatvely effcent plan maxmzes return at pont K. The producton plan that results n the rsk-return combnaton at pont A s less proftable and, thus, s ether allocatvely or techncally neffcent or both. The plan that yelds pont B s also neffcent by reference to pont K; however, t s effcent n the sense that t acheves the maxmum feasble return at the level of rsk, OC. V.A. Return and Rsk Effcency If the measure of effcency s amended to allow managers to trade return for reduced rsk, t can be gauged as a dstance from the fronter at ponts other than the maxmum return. We measure effcency n four dfferent ways. Followng Hughes and Moon (1995), return and rsk effcency are measured along the shortest ray from pont A to pont F on the fronter, the orthogonal ray. Orthogonal return effcency s then the rato 1! (HD/OH) whle orthogonal rsk effcency s the rato 1! (CG/OG). Alternatvely, the orthogonal dstance to the fronter from pont A can be used to measure effcency and equals [(OH!OD) +(OC!OG) ] Our fourth effcency measure uses the one-sded error term from the fronter estmaton to separate the bank s luck of the draw from ts nherent neffcency: vertcal return effcency s gven by: 1 & E(u * ) 0 % 1 RK ) % 2 RK ) 2 (13) where E(u * ) ' u v & & (14)

17 denotes the condtonal expectaton of u gven (Jondrow, Lovell, Materov, and Schmdt (1982)). 15 VI. The Data To estmate the mpact of geographc dversfcaton on return and rsk-takng, we examne BHC data taken from the FRY-9 Fnancal Statements for all four quarters of (The data on the number of branches are from Summary of Deposts reports.) We exclude BHCs located n unt bankng states, BHCs that conssted prmarly of nonbank banks, or specal purpose banks. De novos are excluded and are defned as BHCs that were not operatng as of June The fnal sample ncludes a total of 443 BHCs, rangng n sze from $32.5 mllon to $249.7 bllon n consoldated assets. 8 For our producton model, we specfy the quantty and prces for fve outputs (y varables) and fve nputs (x varables), the level of fnancal captal (k), nonperformng assets (n), and nonnterest ncome (m). The varables are measured as averages over the four quarters of Our output measures are lqud assets (y ), short-term securtes (y ), long-term securtes (y ), th loans and leases net of unearned ncome (y 4), and other assets (y 5). The prce or yeld of the output s th measured by the rato of total ncome from the asset dvded by the quantty of that asset that s accrung nterest. Ths s a measure of the contractual nterest rate rather than the ex-post realzed nterest rate, whch wll depend on the level of ex-post defaults. Our nput measures are fnancal captal (k), labor (x ), physcal captal (x ), nsured deposts 1 2 (x ), unnsured deposts (x ) and other borrowed money (x ) and the model also ncludes the assocated nput prces (w,...,w ). 1 5 A unque feature of ths producton model, whch was frst dscussed n HLMM (1995), s ts 9 treatment of output prces as a measure of qualty. The dfferental between the contractual rate of nterest and the rsk-free rate ndcates the average premum ncurred by the output and therefore s an ndcator of the average qualty of the asset. The weghted-average output prce s defned as y p ' j p. j y j j

18 16 The estmated model also uses the amount of nonperformng assets, n, as an addtonal ndcator of a BHC s fnancal condton. Nonperformng assets are measured as the sum of loans, leases, and other assets past due 90 days or more and stll accrung nterest plus nonaccrung loans, leases, and other assets. 10 Actual profts depend on the random realzaton of a stochastc process, and the ex-post realzaton of proft may not be an accurate ndcator of the ex-ante dstrbuton of returns that motvated the producton plan. Snce actual or realzed proft may be qute dfferent from the expected proft that motvated the producton plan (y,x,k), nstead of usng actual earnngs, we use potental revenue as a proxy for expected revenue. Potental revenue s the revenue that would be earned f all assets accrued nterest. Potental revenue s the sum, p@y + m, snce y ncludes all assets, accrung and nonaccrung, whle p measures the contractual nterest rate. Potental accountng proft s p@y + m! w@x. Snce the federal tax rates are smlar for all the BHCs n the data set, the man varatons n tax rates come from the state tax component of p. The state tax rates are obtaned for each state from The Book of the States, publshed by the Councl of State Governments, and from Sgnfcant Aspects of Fscal Federalsm, publshed by the U.S. Advsory Commsson on Intergovernmental Relatons. VII. The Emprcal Fndngs The estmaton of the structural model of producton, equatons (7), (8), and (10), yelds measures of expected proft and the standard error of expected proft, condtoned on the level of equty 11 captal. Dvdng both terms by equty captal, we obtan for each BHC ts expected rate of return on equty and rsk, and estmate a best practce fronter, whch s shown n Fgure The effects of geographc dversfcaton on expected return, rsk, effcency, and bank safety are nvestgated by regressng these performance measures on varables that characterze the degree of geographc dversfcaton. The nverse z-score: 13

19 17 1/z ' S(p ) [k % E(p )] ' S(p )/k [1 % ER] ' RK 1%ER, (15) measures the probablty of nsolvency, whch s nversely related to bank safety. VII.A. Geographc Dversfcaton, Performance, and Safety We characterze a BHC s geographc dversty n several ways. We focus on the BHC's number of branches and the number of states n whch t operates. Frst, controllng for asset sze and the number of states, we examne the effect on performance and safety of the number of branches. Second, controllng for asset sze and the number of branches, we consder how the number of states n whch the BHC operates affects performance and safety. Thrd, we consder how sze, gven the number of states and branches, affects performance. Fourth, we ask how a proportonal varaton n both the number of states and branches, gven asset sze, affects performance. Last, we consder a proportonal varaton n the number of states and branches and n asset sze. Each of these varatons can be expected to mprove dversfcaton. As dscussed n secton I, ther effects on the rsk-return fronter depend on whether banks are effcent or neffcent. Enhanced dversfcaton shfts the bank s rsk-return fronter upward at all postve levels of rsk for neffcent BHCs. Enhanced dversfcaton could change the poston of the BHC on the envelope effcent fronter. In addton to the geographc varables (.e., the number of states and branches and the amount of total assets), the change n total assets from the last quarter of 1993 to the last quarter of 1994 s ncluded to control for the effects of the growth rate. A dummy varable dstngushng one-bank holdng companes s also ncluded. The results of these regressons are reported n Tables 1 and 2. As shown n Table 1, when the number of branches s vared, holdng constant sze and the number of states, there s evdence that effcency s mproved. The effects on both orthogonal measures and the vertcal return measure of effcency are sgnfcantly postve for the neffcent group. An

20 18 ncrease n branches has a sgnfcant negatve effect on rsk and an nsgnfcant effect on expected return. Thus, for the neffcent, a wder branchng network reduces rsk, reduces ts orthogonal dstance from the fronter, and mproves effcency. For the effcent, both rsk and expected return are sgnfcantly reduced by an ncrease n branchng. (In fact, branchng s the only varable that sgnfcantly affects return, rsk, or bank safety for effcent BHCs.) These two effects suggest that more extensve branchng moves the effcent BHC down the envelope toward lower return and rsk. Consstent wth ths nterpretaton, the nverse z-score s sgnfcantly reduced for effcent banks wth more branches. When the number of states s expanded, holdng constant the amount of total assets and the number of branches, the rsk of neffcent BHCs s sgnfcantly ncreased, wth an almost sgnfcant postve effect on expected return. There s no sgnfcant mpact on any of our four measures of effcency. Insolvency rsk, as measured by the nverse z-score, ncreases as the number of states ncreases. On the other hand, an expanson n the number of states does not have a sgnfcant mpact on the expected return, rsk, or nsolvency rsk of effcent BHCs, although the sgns are opposte to those of an ncrease n branches. When total assets are ncreased, holdng constant the number of states and branches, there s no sgnfcant effect on the effcent banks whle, for the neffcent, rsk s sgnfcantly ncreased, effcency by all four measures s sgnfcantly reduced, and bank safety s sgnfcantly dmnshed. A BHC that s geographcally dversfyng mght be expected to ncrease both branches and number of states. Thus, we nvestgate the effect when the number of states and branches s ncreased proportonately, holdng the asset sze of the BHC constant. (Holdng the asset sze constant focuses on the effect of dversfcaton as dstnct from the effect of a change n sze.) Tables 1 and 2 report the average value of ths dervatve for the neffcent and effcent BHC subsamples, respectvely. We fnd that, on average, the expected return of neffcent BHCs s sgnfcantly ncreased by the proportonate ncrease, rsk s nsgnfcantly ncreased, both orthogonal effcency measures and the vertcal return

21 19 effcency are sgnfcantly mproved, and the nverse z-score s nsgnfcantly ncreased. On average, there s no sgnfcant effect on expected return, rsk, or bank safety for effcent BHCs. Ths suggests that once restrctons on geographc dversty are relaxed we would not expect to see effcent BHCs located at any specfc regon on the envelope fronter but rather scattered over the envelope. Fnally, we also nvestgated the effect of a proportonate ncrease n branches, states, and assets. On average, for neffcent BHCs, the effect on expected return s stll sgnfcantly postve, but the effect on rsk becomes sgnfcantly postve, reflectng the effect of asset growth on rsk. These two effects counterbalance each other so there s no longer a sgnfcant effect on effcency. The expanson now mples, on average, a sgnfcant ncrease n an neffcent BHC s nverse z-score,.e., a decrease n bank safety. On average, for effcent BHCs, the expanson stll has an nsgnfcant effect on expected return, rsk, and bank safety. A number of studes have consdered how geographc expanson affects the return and rsk of these BHCs. Although there are substantal dfferences n measures and methods, Rose (1995) ends hs survey of these studes by concludng that most of them fnd lttle evdence that geographc expanson mproves proftablty or reduces rsk. In fact, rsk s often found to ncrease. Chong (1991) uses stock market returns and the event-study methodology and fnds that nterstate expanson ncreases proftablty and rsk. Chong concluded that, whle geographc expanson reduces the rsk of a gven portfolo, t also expands the bank's opportuntes to take rsks. A smlar concluson s reached by Demsetz and Strahan (1995), who examne stock market data and fnd that, when they control for the bank's portfolo composton and actvtes, mproved dversfcaton reduces rsk. When they allow banks to adjust to mproved dversfcaton, rsk s no longer necessarly reduced. Prelmnary fndngs of Akhaven, Berger, and Humphrey (1995) demonstrate that large bank mergers sgnfcantly mprove proft effcency and that ths mprovement seems to result from ncreases n loan-to-asset ratos and ncreases n banks' leverage. They conclude that the larger scale of these merged banks and ther greater geographc dversty allow them to ncrease ther loan-to-asset ratos and leverage wthout ncreasng

22 20 rsk. VIII. Conclusons and Polcy Implcatons Methods and measures dffer between ths study and those cted, but the complex ntutons and conclusons are remarkably smlar. The evdence does not reject the smple ntuton that better geographc dversfcaton leads to mproved proftablty, reduced rsk, and enhanced bank safety. Instead, the evdence qualfes ths often stated presumpton. And the qualfcatons are mportant. The smple ntuton mplctly assumes that rsk s exogenous. The analytcal devce of manageral utlty and the characterzaton of producton plans n terms of ther mpled return and rsk make evdent the essental endogenety of rsk. Hghlghtng the dfferences between effcent and neffcent banks, t also clarfes the effects on producton of mproved dversfcaton and demonstrates that there s not an ex ante predctable relatonshp between endogenous rsk and dversfcaton. Wth ntuton sharpened, the focus can turn to the mportant mplcatons of geographc dversfcaton for bank safety. Our results ndcate that mproved opportuntes for geographc expanson wll yeld ncreased average returns and ncreased effcency n the bankng system, snce a proportonal ncrease n branches and states yelded sgnfcantly postve effects on expected returns and effcency of the neffcent BHCs n our sample, whch make up more than 88% of our sample. Our estmates ndcate that geographc expanson (holdng asset sze constant) does not have a sgnfcant mpact on nsolvency rsk for the BHCs n our sample. What do these results mean for the safety of the bankng system as a whole, the aggregate number of bank falures, and the aggregate expected losses from bank falures? We calculated the weghted aggregate effect on the nverse z-scores across our sample of a proportonate ncrease n the number of states and branches, holdng assets constant, where each BHC s weght s ts share of assets. (Recall that the estmates reported at the bottom of Tables 1 and 2 are the average effect of a proportonate ncrease.) Ths weghted aggregate effect s! (wth standard error and

23 21 t-statstc );.e., an ncrease n expanson mples a decrease n nsolvency rsk. Ths s consstent wth the vew that ncreased geographc expanson s lkely to mprove aggregate bank safety. However, ths statstc alone s not suffcent to determne the drecton of expected nsolvences and expected losses for the bankng system as a whole. Determnng the drecton and magntude of these aggregate effects wll be a subject of future research.

24 22 REFERENCES Akhaven, Jalal D., Allen N. Berger, and Davd Humphrey. The Effects of Megamergers on Effcency and Prces: Evdence from a Bank Proft Functon, manuscrpt, Board of Governors of the Federal Reserve System, December Berger, Allen N., and Robert DeYoung. Problem Loans and Cost Effcency n Commercal Banks, manuscrpt, Board of Governors of the Federal Reserve System, October Blar, Roger D., and Arnold A. Heggestad. Bank Portfolo Regulaton and the Probablty of Falure, Journal of Money, Credt, and Bankng 10 (1978), Chong, Beng Soon. The Effects of Interstate Bankng on Commercal Banks' Rsk and Proftablty, Revew of Economcs and Statstcs (1991), Demsetz, Rebecca S., and Phlp E. Strahan. Dversfcaton, Sze, and Rsk at Bank Holdng Companes, Federal Reserve Bank of New York, Research Paper #9506, Aprl Hancock, Dana. The Fnancal Frm: Producton wth Monetary and Nonmonetary Goods, Journal of Poltcal Economy 93 (1985), Hancock, Dana. A Model of the Fnancal Frm wth Imperfect Asset and Depost Labltes, Journal of Bankng and Fnance 10 (1986), Hughes, Joseph P., Wllam Lang, Loretta J. Mester, and Choon-Geol Moon. Recoverng Technologes that Account for Generalzed Manageral Preferences: An Applcaton to Non-Rsk-Neutral Banks, Workng Paper No. 95-8/R, Federal Reserve Bank of Phladelpha, September Hughes, Joseph P. and Choon-Geol Moon. Measurng Bank Effcency When Managers Trade Return for Reduced Rsk, Workng Paper, Department of Economcs, Rutgers Unversty, September Hughes, Joseph P., Wllam Lang, Loretta J. Mester, and Choon-Geol Moon. Effcent Bankng Under Interstate Branchng, Journal of Money, Credt, and Bankng (forthcomng). Hughes, Joseph P., and Loretta J. Mester. A Qualty and Rsk-Adjusted Cost Functon for Banks:

25 23 Evdence on the 'Too-Bg-to-Fal' Doctrne, Journal of Productvty Analyss 4 (1993), Hughes, Joseph P., and Loretta J. Mester. Bank Captalzaton and Cost: Evdence of Scale Economes n Rsk Management and Sgnalng, Workng Paper No. 96-2, Federal Reserve Bank of Phladelpha, December Jondrow, J., C.A.K. Lovell, I.S. Materov, and P. Schmdt. On the Estmaton of Techncal Ineffcency n the Stochastc Fronter Producton Functon Model, Journal of Econometrcs 19 (1982), Koehn, Mchael, and Anthony M. Santomero. Regulaton of Bank Captal and Portfolo Rsk, Journal of Fnance 35 (1980), McAllster, Patrck H., and Douglas McManus. Resolvng the Scale Effcency Puzzle n Bankng, Journal of Bankng and Fnance 17 (1993), Mester, Loretta J. How Effcent Are Thrd Dstrct Banks? Busness Revew, Federal Reserve Bank of Phladelpha (January/February 1994), Rose, Peter S. Dversfcaton and Interstate Bankng, manuscrpt, Texas A&M Unversty Graduate School of Busness, March 1995.

26 FIGURE 1. Shftng Rsk-Return Fronter 24

27 FIGURE 2. Iso-1/z Contour 25

28 FIGURE 3. Rsk-Return Fronter Condtonal on Equty Captal 26

29 FIGURE 4. Rsk-Return Fronter of U.S. BHCs, 1994 Data 27 Expected Return Actual BHC Fronter Rsk

30 28 TABLE 1 1 INEFFICIENT BHCs Dependent Varable Inverse z-score Expected Return Rsk E(u*v-u) Coeffcent t-stat Coeffcent t-stat Coeffcent t-stat Coeffcent t-stat Constant E E One Bank Holdng Co E E E Number of States E E E E Number of Branches E E E E Change n Assets E E Total Assets E E E E Sample Sze R-squared Proportonal Geographc E E E E Expanson /Holdng Sze Constant 2 Proportonal Geographc E E E E Expanson /Allowng Sze To Vary 1 Bold numbers ndcate sgnfcance at the 10% level or better. 2 The proportonal geographc expanson results are derved estmates from the estmated regresson above.

31 29 TABLE 1, con t. INEFFICIENT BHCs Dependent Varable Orthogonal Rsk Orthogonal Return Orthogonal Dstance to Effcency Effcency the Fronter Coeffcent t-stat Coeffcent t-stat Coeffcent t-stat Constant One Bank Holdng Co Number of States.75929E E Number of Branches.37075E E E Change n Assets Total Assets E E E Sample Sze R-squared Proportonal Geographc Expanson /Holdng Sze Constant Proportonal Geographc E Expanson /Allowng Sze To Vary

32 30 TABLE 2 3 EFFICIENT BHCs Dependent Varable Inverse z-score Expected Return Rsk Coeffcent t-stat Coeffcent t-stat Coeffcent t-stat Constant E One Bank Holdng Co E E E Number of States E E Number of Branches E E E Change n Assets E E Total Assets E E E Sample Sze R-squared Proportonal Geographc E E Expanson /Holdng Sze Constant 4 Proportonal Geographc E E E E Expanson /Allowng Sze To Vary 3 Bold numbers ndcate sgnfcance at the 10% level or better. 4 The proportonal geographc expanson results are derved estmates from the estmated regresson above.

33 31 NOTES 1. See Rose (1995) for a summary of these studes. 2. In ths paper the term bank safety refers to the rsk of nsolvency, whle rsk means the rskness of bank returns. 3. See HLMM (forthcomng) for a more complete dscusson of ths methodology. 4. Gven a rsk-free asset, the concavty of the rsk-return fronter wll occur f the cost of funds s senstve to the bank s level of rsk. 5. Ths assumes that there are effcent BHCs that are not geographcally dversfed, whch s true n our sample. 6. A smlar procedure was used by Blar and Heggestad (1978) to analyze the effects of portfolo regulaton on bank safety. See also Koehn and Santomero (1980). The nverse z-score s an exact measure of the probablty of nsolvency when profts are normally dstrbuted. 7. A suffcent condton for ths result s that the fronter be non-convex. 8. Ths data set s dentcal to the one used n HLMM (forthcomng). See that paper for a complete defnton of the varables used n estmatng the producton model. A summary of the data s avalable from the authors. 9. Note that the endogenety of prces s not here an ndcaton of market power. It s assumed that there s an exogenous market prce for outputs of a gven qualty. 10. Berger and DeYoung (1995) dscuss the varous relatonshps between problem loans and bank effcency. 11. A Wald test of the condtons for rsk neutralty has a p-value close to zero. Hence, rsk neutralty s rejected. 12. Ths graph s dentcal to Fgure 1 n HLMM (1995). The parameter estmates for the producton model and the effcent fronter are avalable from the authors. 13. There s a dfference between the nsolvency rsk of a BHC and that of the ndvdual subsdares. However, several factors te the two together. The cross guarantee provsons of FIRREA (1989)

34 32 make commonly controlled depostory nsttutons lable for any losses. In addton, BHCs can borrow to supply captal to ther bank subsdares.