Financial Stability and Basel II

Transcription

1 Fnancal Stablty and Basel II by Paul H. Kupec July 006 BSTRCT Basel II wll enhance fnancal stablty f t mproves upon the 988 Basel ccord s ablty to ensure that systemcally mportant nsttutons retan adequate mnmum captal. Quanttatve mpact studes (QIS) show that the dvanced Internal Ratngs Based (IRB) approach the Basel II approach that wll be used by the largest nternatonally actve banks requres sgnfcantly less captal than the 988 Basel ccord. QIS results show IRB banks produce wdely dvergent estmates of mnmum captal for substantally dentcal rsks. These results rase questons about IRB calbraton and the regulatons that defne IRB nputs. IRB captal requrements are compared to captal set usng a fully-specfed equlbrum structural model of credt rsk. The comparson shows that a fully complant IRB bank may produce a wde range of captal requrements for a gven level of rsk exposure owng to regulatory ambgutes regardng the measurement of loss gven default and exposure at default. The results show the IRB approach wll undercaptale portfolo credt rsk relatve to the Basel II target of 99.9 percent bank solvency, and captal shortfalls can be substantal. In contrast, the Foundaton Internal Ratngs Based (FIRB) approach allocates sgnfcantly more captal than necessary to acheve the supervsory objectve. These results suggest the potental for ncreased systemc rsk under Basel II, as regulatons specfy the weakest rsk management standards for large complex IRB banks. Dvson of Insurance and Research, Federal Depost Insurance Corporaton. The vews expressed are those of the author and do not reflect the vews of the FDIC. I am grateful to Sudpto Bhattacharya, Charles Calomrs, Sanjv Das, ark Flannery, Charles Goodhart, chael Gordy, Davd Jones, George Kaufman, Robert Jarrow, Wenyng Jangl, Dlp adan, Jack Redhll, and Haluk Unal for comments on an earler draft of ths paper. Emal: pkupec@fdc.gov

2 - - Fnancal Stablty and Basel II. INTRODUCTION Under the June 004 Basel II agreements, natonal supervsory authortes may choose among three alternatve frameworks to set mnmum regulatory captal for ther nternatonally actve banks. One approach, the standarded approach, sets mnmum captal standards usng a modfed verson of the 988 Basel Captal ccord that lnks captal requrements to external credt ratngs. The remanng two approaches, the so-called Foundaton (FIRB) and dvanced (IRB) Internal Ratngs Based approaches use a regulatory model to assgn mnmum captal requrements. The model assgns captal accordng to an ndvdual credt s probablty of default (PD), loss gven default (LGD), maturty (), and expected exposure at default (ED). The Basel II IRB and FIRB frameworks set mnmum regulatory captal requrements usng a modfed verson of an ndustry model, the so-called Gaussan asymptotc sngle rsk factor model of credt rsk. Ths model, orgnally developed by Vascek (99) (hereafter, the Vascek model), has been extended by many ncludng Fnger (999), Schönbucher (000) and Gordy (003). The Vascek model assumes that default rsk s generated by Gaussan uncertanty and ncludes a sngle common source of rsk and ndependent rsk factors for each credt. Loss gven default (LGD) s specfed exogenously. revsed verson of the June 004 Basel II agreement appears n Basel Commttee for Bankng Supervson (006b). In the U.S., bankng supervsors have determned that Basel II mplementaton wll requre only the largest banks, the so-called core banks, to adopt the IRB approach, whle other banks may petton supervsors for IRB captal treatment (socalled opt-n banks). The remanng banks (so-called general banks) wll contnue usng the captal requrements specfed n the 988 Basel Captal ccord. Core banks are defned as nsttutons wth total bankng (and thrft) assets of $50 bllon or more or total on-balancesheet foregn exposure of $0 bllon or more. General banks lkely wll be subject to a modfed verson of the 988 Basel ccord, so-called Basel, but the potental modfcatons have to be fnaled. - -

3 - 3 - The model assumes bank portfolos are fully-dversfed wth respect to dosyncratc sources of rsk, and captal s needed to buffer the loss uncertanty assocated wth the sngle nondversfable source of rsk. Under the IRB rule, banks must estmate each credt s PD, ED and LGD. Under the FIRB rule, LGD s a fxed regulatory parameter. In contrast to the Vascek model, IRB captal rules specfy default correlaton as determnstc functon of PD that dffers accordng to other characterstcs of the credt. 3 In desgnng Basel II, the Basel Commttee on Bankng Supervson (BCBS) specfed a prudental standard of a 99.9 percent bank solvency rate of over a one-year horon. The BCBS defend the framework s complexty and targeted prudental rgor as necessary gven the complexty of large nternatonal bankng organatons and the need to foreclose opportuntes for regulatory arbtrage that exst under the 988 Basel ccord. 4 lternatve Basel II approaches are calbrated so that the IRB approach produces the lowest captal requrements to encourage banks to transton from the Standarded and FIRB approaches to the IRB approach. Captal savngs accorded under the IRB are ntended to offset the costs assocated wth developng and operatng IRB systems and to reflect effcences that are presumed to be generated by more effcent measurement of credt rsk and assgnment of mnmum captal. BCBS dscussons of the Basel II suggest that the new captal standard should fortfy the mnmum bank captal requrements of nternatonally actve banks. In contrast, the most recent Quanttatve Impact Studes (QIS) rules show that large nternatonally actve banks wll beneft from large captal reductons under Basel II, especally under the IRB approach. These QIS studes also show that IRB estmates of mnmum captal requrements for postons wth smlar rsks may vary wdely across banks. The declne and varablty n regulatory captal estmates rases ssues regardng the calbraton of the IRB See Gordy (003) for further dscusson. 3 The correlaton functon dffers for corporate, soveregn and bank, small- and medum-sed enterprses, specaled lendng, commercal real estate, resdental mortgages, credt cards, and other retal exposures. 4 See for example, BCBS (998, June 999), Jones (000), or yer (00)

4 - 4 - model and the regulatons that gude PD, ED, and LGD nputs nto the IRB rule. To date, no publshed study has analyed the rgor of the prudental standards that are set by the June 006 Basel II IRB models or examned the mplcatons of ther calbraton and nput specfcatons. Ths paper analyes the mnmum solvency standards that are set under the June 006 Basel II IRB approaches for corporate, soveregn, and bank credts. 5 Usng a fully parametered equlbrum structural model of credt rsk, captal allocatons are derved for credt portfolos that satsfy the assumptons that underpn the FIRB and IRB models default correlatons are drven by a sngle common factor, and dosyncratc rsk s fullydversfed. In a calbraton exercse, the June 006 IRB specfcatons of the IRB and FIRB captal requrements are compared to captal estmates calculated from a full structural captal allocaton model. The calbraton comparson shows that the IRB approach substantally undercaptales credt rsks, producng a captal shortfall that vares dependng on the defntons of ED and LGD that are used to calculate IRB captal requrements. Under the defntons of ED and LGD that are most lkely to be used, bank default rates may exceed 5 percent when mnmum captalaton rates are fully complant wth Basel II IRB rules. In contrast, the FIRB wll substantally over-captale credt rsks relatve to the 99.9 percent target solvency rate. Overcaptalaton s a result of the FIRB assumpton of 45 percent LGD, an assumpton that overestmates the loss rates on the credts examned n ths study. The analyss dentfes an mportant source of ambguty n the Basel II captal rules. Under the current framework, permssble defntons for ED and LGD can result n IRB prudental standards that are far weaker than those set by the 988 Basel ccord or those mandated under the alternatve Basel II captal regmes. The IRB rule, moreover, s subject to nterpretaton that may lead to substantal varaton n bank estmates of the captal needed for a gven credt rsk exposure so all IRB banks need not face the same prudental captal 5 The June 006 IRB and FIRB calbratons for corporate, soveregn and bank credts are unchanged from the calbraton n the June 004 Basel II agreements

5 - 5 - standard. The FIRB prudental standard, n contrast, s much more conservatve than the IRB standard that wll apply to the largest systemcally mportant banks. Should IRB bank captalaton levels approach the regulatory mnmums allowed under Basel II, the largest nternatonally actve bankng nsttutons wll operate under a solvency standard that ncreases the mplct publc safety net subsdy. The tlted playng feld establshed under Basel II may encourage bankng system assets to mgrate toward IRB banks n order to maxme the value of mplct safety net subsdes that may accrue to large systemcally-mportant banks. Ultmately the captal benefts that accrue to IRB banks benefts that are ntended to encourage adopton of sophstcated rsk management systems may undermne the Basel II objectve of buttressng the stablty of the nternatonal fnancal system. n outlne of ths paper follows. Secton supples background on the development and calbraton of Basel II. Secton 3 summares the general methodology for constructng economc captal allocatons. Secton 4 derves captal allocatons for portfolo credt rsk n the context of an equlbrum structural model. Secton 5 derves equlbrum credt rsk captal rules under assumptons that mmc those that underle the Basel II IRB models. Secton 6 revews the procedures for settng mnmum captal requrements under the Basel IRB approaches. Secton 7 dscusses the calbraton results and Secton 8 concludes the paper.. BCKGROUND In desgnng ts prudental standards, the Basel Commttee on Bankng Supervson (BCBS) expressed an objectve that ncludes settng mnmum regulatory captal requrements at a level consstent wth a mnmum bank solvency margn of 99.9 percent over a one-year horon. 6 The ratonal for settng ths solvency standard has not been explaned n offcal BCBS documents, and yet nfluental BCBS member studes have 6 Basel Commttee on Bankng Supervson, 004, paragraph

6 - 6 - suggested that ths solvency standard s approxmately consstent wth the standard set by the 988 Basel ccord. 7 The BCBS arrved at the June 004 IRB framework through a process of ndustry consultaton and nternatonal compromse. There s no BCBS document that provdes formal back testng evdence of the solvency standards that are acheved under the alternatve Basel II regmes. Instead, a seres of techncal workng papers and an early draft of the proposed framework were released by the BCBS for publc comment. seres of three Quanttatve Impact Studes (QIS) studes requred banks to estmate the effects of alternatve IRB calbratons on ther mnmum regulatory captal requrements. The June 004 calbratons reflect an teratve process n whch consecutve IRB formulatons were modfed followng ndustry commentary, nternal BCBS dscussons and negotatons, and revew of the QIS results. Calbratons were modfed wth a goal of achevng captal neutralty whle creatng ncentves that encouraged banks to adopt the IRB approaches. 8 Followng the June 004 publcaton of the Basel II framework, two addtonal QIS studes have been conducted: QIS 4 n the Unted States, Germany and South frca, and QIS 5 n adoptng countres n the remander of the world. Both studes reported substantal declnes n mnmum captal requrements for IRB banks relatve to requred captal under the 988 Basel ccord. The Sprng 005 QIS 4 study ncluded 6 U.S. nsttutons, all of whch reported usng the IRB approach. 9 The results show that, n aggregate, mnmum regulatory captal for these nsttutons fell by 5.5 percent under the IRB. mong these banks, the medan reducton n captal was 6 percent and the medan reducton n Ter I captal requrements was 3 percent. Of the few banks that experenced ncreases n mnmum captal requrements under the IRB, the ncreases were drven prmarly by ncreases n captal for consumer retal portfolos and to a lesser extent by equty exposures. 7 See for example Jackson, Perraudn, and Saporta (00). 8 The Basel Commttee on Bankng Supervson, October 00, paragraphs See the Federal Reserve Board Press release, Summary Fndngs of the Fourth Quanttatve Impact Study, avalable at

7 - 7 - In addtonal to large declnes n captal, QIS 4 results show a hgh degree of dsperson n reported estmates of mnmum captal requrements. Banks reported wdely dvergent captal estmates for ther consttuent portfolos (corporate, SE, etc.). Whle these dfferences could owe to true dfference n bank rsk profles as a result of dfferentaton among customer bases and busness strateges, addtonal analyss usng shared natonal credt data ndcated that banks reported wdely dvergent captal estmates for postons wth substantally smlar rsk characterstcs. Further analyss suggests that a sgnfcant share of the varaton n QIS 4 results may be attrbuted dfferences n bank estmates of PDs and LGDs among credts wth approxmately equvalent rsk characterstcs. For the corporate, soveregn, and bank credt portfolo, for example, QIS 4 LGD estmates on non defaulted credts vared from about 5 to 55 percent across bankng nsttutons. The Sprng 006 QIS 5 study ncluded 38 banks n 3 countres outsde of the U.S.. 0 Of the banks that partcpated, the largest nternatonally actve banks, so-called Group banks, posted captal declnes of 7. percent on average under the IRB approach. Smaller banks, so called Group banks, prmarly natonally focused nsttutons, posted much larger declnes n mnmum regulatory captal. Wthn Europe, Group banks posted average captal declnes of 8.3 percent under the IRB. For European Group banks, declnes averaged 6.6 percent under the IRB. Of the banks that experenced large declnes n mnmum regulatory captal requrements, the declnes were attrbuted to bank concentratons n retal lendng, especally resdental mortgages. The BCBS summary of QIS 5 results does not provde a very detaled analyss of the dsperson of bank mnmum captal estmates. The study does however report sgnfcant varaton n IRB nput values. 0 See, BCBS (006a). QIS 5 IRB captal rules nclude a.06 scalng factor that was not ncluded n the June 004 calbraton or the nstructons that guded QIS 4. The ncluson of ths scalng factor means the reported captal declnes wll appear less severe than those reported n the U.S.. BCBS (006a). So-called CEBS (Commttee of European Bank Supervsors) banks

8 - 8 - LGD estmates for wholesale credts, for example, range from 0.8 to 67.6 percent across reportng banks. 3. REGULTORY CPITL FOR CREDIT RISKS Basel II s desgned to ensure a one-year bank solvency margn of 99.9 percent; that s, Basel II mnmum captal requrements ostensbly are formulated to ensure that over a one-year horon, the probablty that a complant bank defaults on ts fnancal oblgatons s less than 0.00 percent. To model the Basel II captal constrant, t s assumed that captal allocatons are set wth a goal of maxmng leverage subject to ensurng that the bank s able to meet all the assocated nterest and prncpal payments wth a mnmum probablty of α = percent. To maxme leverage, the bank s assumed to ssue a sngle class of dscount fundng debt that matures at tme T = year. α = 99.9 percent s the bank s target solvency rate; α = 0.00 percent s the bank s ex ante target probablty of default. The one-year solvency standard mples a captal allocaton horon of one year. The purchased asset, has an ntal market value 0, and a tme random value of wth a cumulatve densty functon Ψ (, ), and a probablty densty functon ψ (, ). 3 Let Ψ (, α ) represent the nverse of the cumulatve densty functon of evaluated at α, α [ 0, ]. Defne an α coverage VaR measure, VaR ( α ), as, VaR α = 0 Ψ, α () ( α ) ( ) ( ) VaR s the loss amount that s exceeded by at most ( α ) of all potental future value realatons of. Expresson () measures value-at-rsk relatve to the ntal market value 3 The constructon of an optmal economc captal allocaton s smplfed when portfolos are composed of assets wth non-negatve market values. For purposes of ths analyss, portfolo composton s restrcted to nclude only long postons n fxed ncome clams that may generate losses that are bounded above by the ntal market value of the credt. See Kupec (004a) for further dscusson

9 - 9 - of the asset. When credt rsk losses are bounded above by below by 0. Ψ (, α ) 0, s bounded Consder a captal allocaton rule that sets equty captal equal to VaR ( α ). By defnton, the probablty the bank wll experence a loss n excess of ts ntal equty value s at most 00 ( α ) percent. Under the ( α ) VaR captal allocaton rule, the bank must borrow VaR ( ) to fnance the nvestment. If the bank borrows VaR ( ), t must 0 α 0 α promse to pay back more than VaR ( ) f equlbrum nterest rates and credt rsk 0 α compensaton are postve. Because the VaR ( α ) captal allocaton rule gnores tme and the equlbrum returns that are requred by bank credtors, the probablty that the bank wll default on ts fundng debt s greater than α f the bank s debts can only be satsfed by rasng funds through the sale of at tme T. The captal allocaton rule that meets the Basel II regulatory objectve s: set equty captal equal to VaR (α ) plus the nterest that wll accrue on the bank s borrowngs. n equvalent allocaton s acheved by settng the par (maturty) value of the fundng debt equal to VaR (α ) and estmatng the fundng debt s market value at ssuance. The dfference between the market value of the purchased asset and the proceeds from the fundng debt ssue s the equty captal needed to fund the nvestment and satsfy the solvency rate target. Ths captal allocaton rule generales to a portfolo context. 4. BSEL II CPITL REQUIREENTS IN N EQUILIBRIU STRUCTURL ODEL Estmaton of the equlbrum nterest cost on fundng debt requres the use of formal asset prcng models or an emprcal approxmaton to value a bank s fundng debt. In ths analyss, we wll adopt the asset prcng approach. If the rsk-free term structure s flat and a frm ssues only pure dscount bond, and asset values follow geometrc Brownan moton, - 9 -

10 - 0 - under smplfyng assumptons, 4 Black and Scholes (973) and erton (974) (hereafter BS) establsh that the market value of a frm's debt s equal to the dscounted value of the bond s par value (at the rsk free rate), less the market value of a Black-Scholes put opton wrtten on the frm s assets. The put opton has a maturty dentcal to the bond s maturty, and a strke prce equal to the par value of the bond. The BS model can be modfed to produce captal allocaton rules that are fully consstent wth equlbrum condtons and the underlyng structural features that determne the market value of bank debt and equty securtes. To smplfy the dscusson, assume all ssued and purchased debt clams mature n one year. The assumpton that bank-ssued debt matures n one year s a convenence that avods the need to mark-to-market bank fundng labltes at the end of the one-year regulatory horon. The assumpton that all credts mature n one year s consstent wth default mode models of credt rsk n whch credts are modeled as ether fully performng or defaultng at the end of the horon of nterest. The Vascek model, and the Basel II IRB models that are derved from the Vascek model are default mode models. The BS captal allocaton framework can easly accommodate longer maturty clams purchased by the bank wthn the regulatory horon of nterest, but there s a cost n terms of mathematcal complexty that s unnecessary for purposes of ths paper. Ths socalled market-to-market captal allocaton process, where long maturty credts are valued at the end of the captal allocaton horon, s dscussed n Kupec (004a, 004b). Consder a bank whose only asset s a rsky BS dscount bond ssued by an unrelated counterparty that matures at date. Let B 0 represent the ntal market value of ths bond. ssume that the bank wll fund ths bond wth ts own -year dscount debt ssue, and equty. In ths settng, the bank s fundng debt ssue s a compound opton. Let and 4 There are no taxes, transactons are costless, short sales are possble, tradng takes place contnuously, f borrowers and savers have access to the debt market on dentcal rskadjusted terms, and nvestors n asset markets act as perfect compettors

11 - - Par P represent, respectvely, the tme value of the assets that support the dscount debt nvestment and the par value of the purchased bond. Let Par F represent the par value of the dscount bond that s ssued by the bank to fund the nvestment. The end-of-perod cash flows that accrue to the bank s debt holders are, n [ n(, Par P ), Par F ]. () The ntal equlbrum market value of the bank s dscount bond ssue s the dscounted (at the rsk free rate) expected value of the end-of-perod fundng debt cash flows taken wth respect to the equvalent martngale probablty dstrbuton for the assets, q rf E[ n[ n(, ParP ), ParF ] e (3) where q and s a standard normal random varable. 0 e r f + The payoff of the bank s purchased bond s gven by [ Par, ] asset value at date, rsk. Let x, and let Φ( x) 0 e μ + q, (4) n P, where s the, and μ r + λ, where λ s the market prce of = f represent the cumulatve standard normal dstrbuton functon evaluated at Φ ( α ) represent the nverse of ths functon forα [ 0,]. Because s monotonc n, the upper bound on the par (maturty) value of the fundng debt that can be ssued under the target solvency constrant s, ( α ) Ψ (, α ) 0 μ + Φ ( α ) Par F = = e. (5) The ntal market value of ths fundng debt ssue s, ( α ) B B, q rf ( α ) = E[ n[ n(, Par ), Par ( α ) ] e F 0 F 0 P F. (6) and the ntal equty allocaton consstent wth the target solvency rateα, E ( α ) E q rf ( α ) = B E[ n[ n(, Par ), Par ( α ) ] e, s, 0 P F. (7) - -

12 - - In the sngle asset case, when the probablty of default on the purchased bond s less than or equal to ( α ), the bond can be fnanced 00 percent wth bank debt wthout volatng the solvency constrant ( Par (α ) = Par ). When the probablty of default on the F purchased bond exceeds ( α ), captal s requred, and Par α ) < Par. In ths case, expresson (7) mples a dollar captal allocaton, P F ( P E ( α ) = B 0 e c r f ln Φ ( ) ln( ) r ln( ) ln( ) c 0 f 0Φ c 0 r f + (8) where μ + Φ ( α ) ( α) c = ParF = 0 e. Portfolo Captal Except for the need to derve a probablty densty functon for the portfolo s future value dstrbuton under both the physcal and the equvalent martngale measures, the process for settng portfolo captal requrements mrrors the calculatons for a sngle asset. In most cases, credt portfolos do not have densty functons that admt a closed-form expresson for ether the par value of the fundng debt or ts ntal market value. onte Carlo smulaton s often requred to estmate VaR (α ) and the par value of the fundng debt, and prcng the fundng debt may requre numercal evaluaton of a hgh order ntegral. The next secton consders portfolo captal allocaton under BS assumptons when asset prce dynamcs are generated by a sngle common factor and dosyncratc rsk s fully dversfed. These assumptons reduce sgnfcantly the complexty of portfolo captal calculatons. 5. BSEL II CPITL UNDER SYPTOTIC SINGLE FCTOR SSUPTIONS Captal allocaton calculatons are smplfed f a portfolo s well-dversfed and asset values are drven by a sngle common factor n addton to ndvdual dosyncratc factors. Let dw represents a standard Wener process common n all asset prce dynamcs, - -

13 - 3 - and represents an ndependent standard Wener process dosyncratc to the prce dynamcs of asset. ssume that asset prce dynamcs for frm are gven by, dw, dw dw dt d μ + + = (9). 0,., 0, dw dw j dw dw m j j = = = = ρ ρ Under these dynamcs, asset prces are log normally dstrbuted, ( ) ( ) T T r T f e 0 λ =, (0) where and are ndependent standard normal random varables. Under the equvalent martngale change of measure, asset values at tme T are dstrbuted, ( ) ( ) T T r q T f e =. () Under these prce dynamcs, the correlaton between geometrc asset returns s, ( ) ( ).,, ln, ln T 0 0 j T Corr j j jt t + + = () If the model s further specaled so that the volatltes of assets dosyncratc factors are assumed dentcal,,,, j j = = the par-wse asset return correlatons are,., ln, ln 0 0 j T T Corr j jt t + = = ρ (3) sset Return Dstrbutons The T-perod return on BS rsky bond that s held to ts maturty date T s, - 3 -

14 - 4 - T ( n(, Par ) T = T. (4) B 0 s bounded n the nterval[,a], where a s a fnte constant. When return realatons are n the range, < < 0, T T represents the loss rate on the bond held to maturty. For realatons n the range, 0< T < Par B 0, the bond has defaulted on ts promsed payment terms, but the bond has stll generated a postve return. fully performng bond posts a return equal to Par B 0 < a whch s fnte by assumpton (and n practce). bond s physcal return dstrbuton (7) has an assocated equvalent martngale return dstrbuton, q ( n(, Par ) q T = T. (5) B By constructon, expressons (4) and (5) have dentcal support. 0 symptotc Portfolo Return Dstrbuton The T-perod return on a portfolo of n rsky ndvdual credts, P T, s P T n = n = T B B 0 0 (6) Let ( P ) P T = = represent the portfolo return condtonal on a realaton T of the common market factor, = m, - 4 -

15 - 5 - P T = n = If ψ ( ) T represents the condtonal return densty functon, under the sngle common factor assumpton for asset prce dynamcs, ψ ( ) T and ( jt ) T n = B 0 B 0 (7) ψ are ndependent for 5 j. Consder a portfolo composed of equal nvestments n ndvdual bonds that share dentcal ex ante credt rsk profles. That s, assume that the bonds n the portfolo are dentcal regardng par value { Par = Par,, j}, maturty {T }, and volatlty j characterstcs, { = =,, j}. Condtonal bond returns are ndependent and j dentcally dstrbuted wth a fnte mean. s the number of bonds n the portfolo, wthout bound, the Strong Law of Large Numbers requres, Lm n n T = [ P T ] = Lm E[ ψ ( T )] n n a. s N, grows (8) The notaton a.s. ndcates almost sure convergence (convergence wth probablty one). Under the BS sngle factor assumptons, expresson (8) becomes, 5 Independence n ths non-gaussan settng requres that an observaton of the return to bond j be unnformatve regardng the condtonal dstrbuton functon for bond, Pr < a = Pr < a gven that =, a, j Ths condton s ( ) ( ) ). t t satsfed under the sngle common factor model assumpton. jt jt - 5 -

16 - 6 - where, Lm n P T = Q( + B 0 Par B ) 0 [ Φ( w ( ))] [ Φ( w ( ) + γ )] T T T (9) μ T 0 f + ( ) = ln [ ] + r + λ ( + ) T T γ T = T w T ( ) = ln[ Par ] μ γ T T ( ) Q( ) = e T T ( γ μ ) + Smlarly, the equvalent martngale portfolo condtonal return dstrbuton s gven by, Lm n P q T Par = B0 q Q ( ) + B 0 q [ Φ( w ( ))] T q [ Φ( w ( ) + γ )] T T (0) where, q μ T 0 f + ( ) = ln [ ] + r ( + ) T T γ T = T w q T ( ) = ln[ Par ] μ γ T q T ( ) - 6 -

17 - 7 - Q q ( ) = e q T T ( γ μ ) + Optmal Portfolo Captal llocaton n a Default ode pproach In the default mode approach used n Basel II, credts are assumed to be held to maturty and ether they fully perform or default. The hold-to-maturty assumpton removes the need to mark portfolo credts to market at a date pror to maturty. In the approach that follows, t s assumed that all credts have an dentcal maturty of T = years consstent wth the Basel II framework. The portfolo return dstrbuton s monotonc n, so the captal allocaton calculatons need only nvolve the condtonal portfolo return dstrbutons. When expressed as a proporton of the nvestment portfolo s ntal market value, the optmal par value of fundng debt, P ( α ) = Ψ ( +, α ) parf P can be determned by settng = Φ ( α ) and usng expresson (9) to solve for the end-of-horon portfolo crtcal value, par P F ( α ) Par B0 = Q( + B [ Φ( w ( Φ ( α ) )] 0 ) [ ( ( ( )) )] Φ w Φ α + γ () To determne the market value of the fundng debt, t s necessary to solve for the value of that determnes the default threshold under the rsk neutral measure, ˆ, ˆ = Φ ( α ) + λ () - 7 -

18 - 8 - ẑ s one of the lmts of ntegraton needed to calculate the expected dscounted payoff of the fundng debt usng the rsk neutral measure. Expressed as a proporton of the nvestment ( α ) portfolo s ntal market value, the ntal market value of the fundng ssue,, s, P bf 0 b P F 0 ( α ) ˆ Par B0 q r ˆ Q ( f = e + B0 P + parf q [ Φ( w ( ))] φ( ) ) ( α )[ Φ( ˆ )] q [ Φ( w ( ) + γ )] φ( ) d d (3) The economc captal allocaton for the portfolo, expressed as a proporton of the portfolo s ntal market value, P K BS ( α ) s, K P BS ( α ) ˆ rf = e + Par B0 ˆ + par q [ Φ( w ( ))] φ( ) q Q ( B P F 0 ) q [ Φ( w ( ) + γ )] φ( ) ( α )[ Φ( ˆ )] d d (4) ( ) P The dollar value captal requred s B 0 K BS α. n = Idosyncratc rsk s fully dversfed, so when an addtonal credt s added to the portfolo, the margnal captal requred to mantan the target solvency margn s equal to the portfolo s average captalaton rate (expresson (4)) multpled by the market value of the margnal credt added to the portfolo. Expresson (4) represents the mnmum regulatory captalaton rate for both the average and the margnal credt n an asymptotc portfolo - 8 -

19 - 9 - when credt rsks are prced to satsfy BS equlbrum condtons and captal s set to acheve a 99.9 percent solvency rate. 6. INIU CPITL REQUIREENTS UNDER THE BSEL II IRB PPROCHES The June 006 formula for calculatng IRB captal requrements for corporate, soveregn and bank exposures are ED K, where K, s gven by 6 K = LGD Φ Φ R R R ( PD) + Φ (.999) (.5) + PD LGD.5b b (5) where, R e 0. e 50 PD e e 50 PD = 50 50, b ( Ln( PD) ) =, PD s a credt s probablty of default expressed as a percentage, LGD s a credt s expected loss gven default expressed as a percentage, s the credt s maturty n measured n years, and K represents the percentage captal requrement per dollar of ED exposure. When =, the maturty adjustment, (.5) +.5b b =. If for any credt, K<0, regulatory captal requrements are set to ero. FIRB captal requrements are calculated by usng the IRB captal requrement formula wth LGD set at 45 percent. 6 BCBS (June 006b), page

20 - 0 - Basel II IRB odel Inputs Probablty of default (PD) For corporate and bank exposures, PD s the greater of the one-year PD or 0.03%. Basel II specfes that PD must be estmated usng at least 5 years of data but for purposes of ths analyss, PD s well defned by the BS asset prce dynamcs and data estmaton requrements are not an ssue. Exposure at default (ED) ccordng to paragraph 474 of BCBS (June 006b) ED for an on-balance sheet or off-balance sheet tem s defned as the expected gross exposure of the faclty upon default of the oblgor. For on-balance sheet tems, banks must estmate ED at no less than the current drawn amount, subject to recognng the effects of on-balance sheet nettng as specfed n the foundaton approach. The U.S. Basel II NPR (p.3) states, ED for the on-balance sheet component of a wholesale or retal exposure means () the bank s carryng value for the exposure (ncludng accrued but unpad nterest and fees) Thus, for smple loans or bonds wthout any addtonal attached lne of credt, Basel II requres that ED must be at least as large as the current carryng value of the asset at the tme that mnmum regulatory captal s calculated. For smple loan or bond postons, Basel II regulatons do not nclude any further dscusson; n partcular, there s no dscusson or gudance that suggests that ED must be hgher than the current carryng value of a smple fully drawn loan. Loss gven default (LGD) Paragraph 97 of BCBS (June 006b) requres that, LGD be measured as a percentage of the ED. Paragraph 468 dscusses mnmum requrements for IRB bank LGD treatment: - 0 -

21 - - bank must estmate an LGD for each faclty that ams to reflect economc downturn condtons where necessary to capture the relevant rsks. Ths LGD cannot be less than the long-run default-weghted average loss rate gven default calculated based on the average economc loss of all observed defaults wthn the data source for that type of faclty. The Basel II gudelnes also do not set a lower bound on the LGDs that banks can use n the IRB approach for corporate credts. aturty () BCBS (June 006b) paragraph 30 defnes as, the greater of one year and the remanng effectve maturty n years, where the remanng effectve maturty s the tme t CFt weghted-average of the nstrument s cash flows, t n, 5 years, where CF t s the CFt t nstrument cash flow t perods nto the future, where t s measured n years. Table : Calbraton ssumptons rsk free rate r =. 05 f market prce of rsk λ =. 0 market factor volatlty =. 0 Frm specfc volatlty =. 0 Intal market value of assets = 0 00 correlaton between asset returns ρ =

22 IPLIED BNK DEFULT RISK UNDER THE BSEL II IRB PPROCHES The economc captal allocatons prescrbed by the BS model are compared to the mnmum regulatory captal requrements that are set by the Basle II IRB approaches. Portfolo captal requrements are calculated usng the BS model and the Basel IRB approaches for portfolos wth a wde range of rsk characterstcs. The assumptons regardng asset prce dynamcs that are mantaned throughout the analyss appear n Table. ll ndvdual credts are assumed to have dentcal frm specfc rsk factor volatltes of 0 percent. The common factor has a volatlty of 0 percent and the market prce of rsk s set at 0 percent. The rsk free rate s 5 percent. The market and frm specfc factor volatltes mply an underlyng geometrc asset return correlaton of 0 percent. 7 ll credts n these asymptotc portfolos are assumed to have the same ntal value, and all share an dentcal ex ante credt rsk profle that s determned by the par value of the credt. The par values of ndvdual credts are altered to change the credt rsk characterstcs of a portfolo. Consstent wth Basel II requrements, the analyss focuses on a one-year captal allocaton horon and one-year maturty credts. In the comparson, the Basel II mnmum captal calculatons nclude expected loss snce there s no scope for loan loss reserves n ths one-perod settng. The modfed Basel II defnton of K s, K = LGD Φ Φ R R R ( PD) + Φ (.999) (.5) +.5b b (6) 7 When the bond PDs n Table are nput nto the -IRB approach, the correlaton parameter, R, ranges from 3.6 (par value 70) to.7 percent (par value 55). - -

23 - 3 - Input Values for the Basel II Internal Ratngs Based pproaches PD The value of a BS bond may vary over ts lfe, but t may only default at maturty. Under the mantaned stochastc assumptons, the physcal probablty that a BS bond defaults at maturty s, = ( ) where, PD Φ df df Log = ( Par ) Log( ) 0 + r + f λ + (7) Whle Basel II regulatons dscuss data requrements for estmatng a -year PD rate from hstorcal data, there s no ambguty about the defnton of PD for the BS bonds used n ths analyss. LGD and ED The expected value of the bond s payoff at maturty, gven that the bond defaults s, E [ n(, Par ) < Par ] = Φ df ( ) df 0 e r + λ f ( + ) + ( + ) φ()d (8) Expresson (8) can be used to defne loss gven default, but there s sgnfcant lattude as to how loss gven default mght be measured under Basel II. Recall that Basel II requres that LGD be measure relatve to ED, whch must be at least as large as current gross exposure. When LGD s calculated as a loss relatve to a credt s current exposure (ntal market value), CE LGD, s, - 3 -

24 - 4 - LGD CE df ( ) ( + ) + ( + ) df f λ e = φ()d B Φ 0. (9) 0 r + Expresson (9) s a current exposure measure of calculatng loss gven default because t measures the default loss relatve to the current exposure at the tme that the captal allocaton s beng estmated. nother potental way to measure loss gven default usng expresson (8) s to measure LGD as the shortfall relatve to the promsed maturty value should a credt default. Defne the future exposure measure of loss gven default, FE LGD, relatve to the full contract value at the end of the captal allocaton horon, as loss s measured LGD FE df ( ) ( + ) T + T ( + ) df rf + λ e = φ( )d Par Φ 0 (30) Clearly LGD > LGD. Wthn the context of the Vascek model and the Basel II framework, there s no theoretcal preference for ether measure of LGD as these models deal exclusvely wth the default process and are slent on the losses generated n default. Indeed there may be other methods for constructng LGD that are wdely appled n practce. s the current exposure method for calculatng LGD s clearly acceptable approach under Basel II, and snce FE CE LGD CE also produces the lowest mnmum regulatory captal requrements, t s reasonable to assume that CE LGD wll be the preferred defnton of IRB banks. lternatve nmum Captal Estmates The analyss ncludes 6 portfolos of one-year credts. The credt rsk characterstcs of the ndvdual exposures are reported n Table. Indvdual credt PDs range from 3-4 -

25 - 5 - bass ponts for a bond wth par values of 55, to 3.99 percent for a bond wth a par value of CE 70. The LGD characterstcs (measured from ntal market value) range from.40 percent to 3.8 percent. When LGD s measured on a future value bass, LGD ranges from 6. to 8.34 percent. Whle the BS model produces only modest LGDs relatve to hstorcal estmates of default losses on rated corporate bonds, the IRB rule explctly accounts for LGD, so a pror, there s no reason to expect that any specfc set of LGD values may compromse the performance of the IRB approach. FV Table : Credt Rsk Characterstcs of -Year Credts expected n percent ntal probablty value loss gven loss gven yeld par market of default gven default from default from to value value n percent default ntal value par value maturty The results of the captal comparson are reported n Table 3 and plotted n Fgure. Dependng on how ED and LGD are defned, IRB captal requrements may take on a - 5 -

26 - 6 - wde range of values. Whle the future exposure defnton of ED and LGD produce the largest IRB captal measures, these captal assgnments stll fall short of the captal needed to acheve a 99.9 percent solvency rate for all but the safest credts analyed. When captal requrements are calculated usng the IRB approach and the current exposure measure of LGD, the true captal needed to acheve the 99.9 percent target solvency rate may be more than 5 tmes larger than the mnmum captal set by the IRB approach. Table 3: Estmates of Portfolo Captal Requrements for -year Credts FIRB FIRB IRB IRB captal captal captal captal 99.9 percent current future current future bond probablty BS exposure exposure exposure exposure par of default captal ED ED LGD, ED LGD,ED value n percent n percent n percent n percent n percent n percent Characterstcs of ndvdual bonds are reported n Table. The Basel FIRB and IRB captal requrements nclude expcted losses as well as unexpected losses. s Fgure shows, owng to the 45 percent LGD assumpton, the FIRB approach dramatcally ncreases captal requrements relatve to the IRB approach. For credts wth probabltes of default less than about.6 percent, FIRB captal requrements provde relef relatve the 8 percent captal requred by the 988 Basel ccord. Notwthstandng captal - 6 -

27 - 7 - reductons for some credts, for the portfolos examned n ths analyss, the FIRB wll set captal requrements that are many tmes larger than are needed to acheve the regulatory target default rate of 0. percent. Under the FIRB approach, use of the future exposure ED measure further compounds the captal surplus. Fgure : Comparson of lternatve nmum Captal Estmates for -Year Credts FIRB future exposure mnmum captal (n percent) FIRB current exposure ED BS IRB future exposure LGD, ED IRB current exposure LGD, probablty of default (n percent) The BS captal allocaton rule can be nverted to recover the mpled probablty of default under the IRB rule. Fgure plots the solvency margns that are set by the IRB framework when captal s set usng both the current exposure and the future exposures measures for ED and LGD. The solvency margn set by the IRB rule depends on the rsk attrbutes of the credt and the - 7 -

28 - 8 - defntons used for the ED and LGD nputs. For the credts examned n ths analyss, usng the current exposure measure of ED and LGD, the IRB solvency margn ranges from 97.5 percent to 93.4 percent. Under ether measure of ED and LGD, the IRB solvency margn declnes as credt rsk ncreases. probablty of default (n percent) Fgure :pproxmate Solvency argns Under Basel II and lternatve LGD easures IRB solvency margn usng future exposure defnton of ED, LGD solvency margn (n percent) 99.9 percent IRB solvency margn usng current exposure defnton of ED, LGD Dscusson No publshed study has attempted to measure the rgor of the prudental standard that wll be set by Basel II captal regulatons. Notwthstandng the BCBS s stated goal of creatng a prudental standard consstent wth a 99.9 percent bank solvency rate, the Basel II Quanttatve Impact Studes (QIS) and subsequent IRB model calbraton adjustments have not focused on producng calbratons consstent wth any specfc target solvency margn. Basel II delberatons have produced IRB model calbratons that create ncentves to - 8 -

29 - 9 - promote IRB adopton, but at the cost of sgnfcant dmnuton n the prudental standards that wll apply to credt rsk exposures n IRB banks. To gan perspectve of the solvency standard set under the IRB approach, consder for a moment that the IRB approach default rates plotted n Fgure are approxmately equal to the falure rate experenced by U.S. savngs and loan nsttutons durng the heght of the 980s S&L crss. In 988, the falure rate among nsured savngs and loans was 6.4 percent. 8 Over the perod, the annual compound average default rate of banks nsured by the Federal Depost Insurance Corporaton was less than. percent, and even n the worst year (988) of a perod that has been charactered as a bankng crss, the default rate on FDIC nsured banks never exceed percent. 9 The use of the BS model as a benchmark of comparson merts dscusson because the model has well-known emprcal shortcomngs. Econometrc studes suggest that, on average, the BS model overprces corporate bonds (.e., underestmates requred bond yelds). Emprcal evdence ndcates that the BS bas s related to maturty and credt qualty. 0 BS overprcng errors are more severe on short-term hgh qualty credts. In the context of ths captal calbraton exercse, the observed pattern of bas mples that the BS model analyss wll lkely understate the true amount of captal that s requred to support a credt rsky portfolo because the bank s fundng debt s lkely to be overprced by the BS 8 Hstory of the Eghtes Lessons for the Future, p See, Hstory of the Eghtes Lessons for the Future, p See for example, Jones, ason, and Rosenfeld (984), Ogden (987), or Eom, Helwege and Huang (004)

30 framework. Recognng the shortcomngs of the BS model, true economc captal allocatons are lkely larger than the estmates n ths paper suggest, and true IRB captal shortfalls are lkely more severe than ndcated. To the extent that banks enjoy safety-net engendered subsdes that are attenuated by mnmum regulatory captal requrements, the Base II IRB calbratons engender ncentves that wll encourage bankng system assets to mgrate toward IRB banks. sset mgraton could be acheved through consoldaton or through an ncrease n the number banks that are granted regulatory approval for the IRB approach. If regulatory hurdles and the fxed costs assocated wth adoptng IRB compatble systems are hgh, only the largest banks wll favor the IRB approach. bsent lberal regulatory approval polces or declnes n IRB complant data and systems, strong economc ncentves are n place to encourage ndustry consoldaton nto nsttutons that gan IRB regulatory approval. graton of assets nto IRB banks could substantally ncrease systemc rsk n the fnancal system as even fully complant IRB banks may have hgh default rates unless market dscplne of other regulatons prohbt the realaton of the full captal relef granted by the IRB approach. For example, the leverage constrant under the U.S. system of prompt correctve acton ( U.S.C. Secton 83) may become the bndng mnmum regulatory captal requrement for many U.S. IRB banks. See for example, the testmony of Donald E. Powell, Charman Federal Depost Insurance Corporaton, on the Development of the New Basel Captal ccords before the Commttee on Bankng, Housng, and Urban ffars. November 0, 005, avalable at:

31 CONCLUSIONS Compared to captal requrements calculated usng a full equlbrum structural model of credt rsk, the Basel II IRB approach substantally understates the captal that s requred to acheve the regulatory target of a 99.9 percent bank solvency rate. Estmates suggest that IRB banks may have default rates n excess of 5 percent on ther corporate soveregn and bank credt portfolos and stll meet the mnmum rsk-based regulatory captal requrements promulgated by the June 006 the IRB approach. In contrast, the FIRB approach requres far more captal than s necessary to meet the regulatory target solvency standard. The captal allocaton analyss n ths paper hghlghts mportant ambgutes n the Basel II framework and these ambgutes may lead to sgnfcant varaton n the captal standards that apply across IRB banks. Current Basel II regulatons are unnecessarly vague regardng the defntons of exposure at default and loss gven default. Dfferent nterpretatons of these concepts can lead to vastly dfferent mnmum regulatory captal requrements under the IRB approach. Regardless of the defntons employed, the IRB framework must stll be recalbrated to produce an ncrease n mnmum captal requrements f the 99.9 percent solvency margn target s to be respected. The recalbraton of the IRB rule must be lnked to reformulated defntons of ED and LGD that remove ambguty and frmly establsh the characterstcs of the nputs used n the IRB captal rule. Under the current Basel II formulaton, banks that adopt the IRB approach wll gan substantal regulatory captal relef wthout a commensurate reducton n ther potental rsk profle. FIRB banks, n contrast, wll face a much strcter prudental standard. Ths dchotomy creates strong economc ncentves for bankng system assets to mgrate nto IRB banks. Snce the analyss suggests that IRB banks potentally carry hgher default - 3 -

32 - 3 - rsk absent safety net support, the mgraton of bankng system assets toward IRB regulatory captal treatment s unlkely to enhance fnancal stablty. Gven the prudental weaknesses assocated wth the IRB approach, the adopton of Basel II n ts current form need not promote better rsk management practces n banks or reduce systemc rsk n the nternatonal bankng system

33 REFERENCES Basel Commttee on Bankng Supervson (004). Basel II: nternatonal Convergence of Captal easurement and Captal Standards: Revsed Framework. Bank for Internatonal Settlements, June. valable at Basel Commttee on Bankng Supervson (00). Overvew Paper for the Impact Study, Bank for Internatonal Settlements, October. valable at Basel Commttee on Bankng Supervson (00). The Internal Ratngs-Based pproach Consultatve Document. Bank for Internatonal Settlements, ay. valable at Basel Commttee on Bankng Supervson (999). Captal Requrements and Bank Behavor: The Impact of the Basel ccord, BCBS Workng Paper No., Bank for Internatonal Settlements, valable at Basel Commttee on Bankng Supervson (006a). Results of the ffth quanttatve mpact study (QIS 5). Bank for Internatonal Settlements, June 6. valable at Basel Commttee on Bankng Supervson (006b). Internatonal Convergence of Captal easurement and Captal Standards: Revsed Framework Comprehensve Verson. Bank for Internatonal Settlements, June 6. valable at Black, F. and J.C. Cox (976). Valung Corporate Securtes: Some Effects of Bond Indenture Provsons, Journal of Fnance, Vol. 3, pp Black, F. and. Scholes (973). The prcng of optons and corporate labltes, The Journal of Poltcal Economy, Vol. 8, pp Eom, Young, Jean Helwege and Jng-h Huang, (004). Structural odels of Corporate Bond Prcng: n Emprcal nalyss, Revew of Fnancal Studes, Vol. 7, pp Gordy, chael (003). rsk-factor model foundaton for ratngs-based bank captal rules, Journal of Fnancal Intermedaton, Vol., pp Greenspan, lan, (998). Remarks by Charman Greenspan before the Conference on Captal Regulaton n the st Century, Federal Reserve Bank of New York, February 6, 998. Fnger, Chrs (999). Condtonal approaches for Credtetrcs portfolo dstrbutons, Credtetrcs ontor, pp Notce of proposed rule makng to mplement Basel II rsk-based captal requrements n the Unted States for large nternatonally actve bankng organatons, arch 30, 006. avalable at

34 Jackson P., W. Perraudn and V Saporta, (00). Regulatory and economc solvency standards for nternatonal actve banks, Journal of Bankng and Fnance Vol. 6, pp Jones, Davd (000). Emergng problems wth the Basel Captal ccord: Regulatory captal arbtrage and related ssues, Journal of Bankng and Fnance, Vol. 4, - (January), pp Jones, E. P., S. ason, and E. Rosenfeld (984). Contngent Clams nalyss of Corporate Captal Structures: n Emprcal Investgaton, Journal of Fnance, Vol.39, pp Hstory of the Eghtes Lessons for the Future. Volume I: n Examnaton of the Bankng Crss of the 980s and Early 990s. Washngton D.C.: Federal Depost Insurance Corporaton, 997. Kupec, Paul (004a). Estmatng economc captal allocatons for market and credt rsks, Journal of Rsk, Vol. 6, No. 4. Kupec, Paul (004b). Captal dequacy and Basel II, FDIC CFR Workng paper No , erton, Robert (974). On the prcng of corporate debt: The rsk structure of nterest rates, The Journal of Fnance, Vol. 9, pp yer, Laurence, (00). Remarks by Governor Laurence H. yer at the Rsk anagement ssocaton s Conference on Captal anagement, Washngton D.C., ay 7, 00. Ogden, J. (987). Determnants of the Ratngs and Yelds on Corporate Bonds: Tests of the Contngent Clams odel, The Journal of Fnancal Research, Vol. 0, pp Schönbucher, P. (000). Factor odels for Portfolo Credt Rsk, memo, Department of Statstcs, Bonn Unversty. Vascek, O.., (99). Lmtng loan loss probablty dstrbuton, KV Corporaton, workng paper