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1 econstor Der Open-Access-Pulikationsserver der ZBW Leiniz-Informationszentrum Wirtschaft The Open Access Pulication Server of the ZBW Leiniz Information Centre for Economics Breuer, Wolfgang; Gürtler, Marc Working Paper Coherent anking capital and optimal credit portfolio structure Working papers // Institut für Finanzwirtschaft, Technische Universität Braunschweig, No. FW2V2 Provided in Cooperation with: Technische Universität Braunschweig, Institute of Finance Suggested Citation: Breuer, Wolfgang; Gürtler, Marc (2006) : Coherent anking capital and optimal credit portfolio structure, Working papers // Institut für Finanzwirtschaft, Technische Universität Braunschweig, No. FW2V2 This Version is availale at: Standard-Nutzungsedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgerauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreien oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insesondere CC-Lizenzen) zur Verfügung gestellt haen sollten, gelten aweichend von diesen Nutzungsedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may e saved and copied for your personal and scholarly purposes. You are not to copy documents for pulic or commercial purposes, to exhiit the documents pulicly, to make them pulicly availale on the internet, or to distriute or otherwise use the documents in pulic. If the documents have een made availale under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. zw Leiniz-Informationszentrum Wirtschaft Leiniz Information Centre for Economics

2 Working Paper Series Coherent Banking Capital and Optimal Credit Portfolio Structure y Wolfgang Breuer and Marc Gürtler No.: FW2V2/06 First Draft: This Version: Braunschweig University of Technology Institute for Economics and Business Administration Department of Finance At-Jerusalem-Str. 7 D-3806 Braunschweig

3 Coherent Banking Capital and Optimal Credit Portfolio Structure y Wolfgang Breuer*and Marc Gürtler** Astract. Coherent measures of a ank s whole risk capital imply a structure of a ank s optimal credit portfolio that is independent of its deposits and the expected deposit rate, of expected ankruptcy costs and of expected costs of regulatory capital. Keywords: Basel II, Regulatory Capital, Coherent Risk Capital, Separation JEL classification: G2, G28 * Professor Dr. Wolfgang Breuer ** Professor Dr. Marc Gürtler RWTH Aachen University Braunschweig University of Technology Department of Finance Department of Finance Templergraen 64, Aachen, Germany At-Jerusalem-Str. 7, 3806 Braunschweig, Germany Phone: Fax: Phone: Fax: wolfgang.reuer@rwth-aachen.de marc.guertler@tu-s.de

4 . Introduction In almost every developed country there are compulsory anking capital requirements that aim at sustaining the staility of the financial sector. According to the new capital adequacy framework (Basel II) finally adopted y the Basel Committee in June 2004 a ank s required regulatory capital (RC) is ased on so-called economic capital descriing that capital amount that a firm elieves is necessary to asor potential losses associated with each of the included risks (Basel Committee on Banking Supervision, 2003, p. ). But since anks are legally oliged to make provisions against expected losses, the capital requirement RC only aims at the prolem of unexpected losses (see Kupiec, 2003). We thus may define a ank s required whole risk capital (WRC) as the sum of its economic or regulatory capital and of its expected losses, ecause the latter leads to provisions a ank can only make if it is furnished with enough capital. Mostly, the WRC is calculated as the Value-at-Risk (VaR), although the Value-at-Risk exhiits some well-known unfavorale features. In particular, according to Artzner et al. (997), as a consequence of the missing suadditivity of the VaR, two anks that merge may ceteris parius face higher capital requirements than the sum of their individual requirements efore the merger. Certainly, this finding is counterintuitive and may induce rather adverse incentive effects for anks regarding their decisions on organizational structures. Therefore, with L and L 2 as two uncertain losses from two different credits and 2, Artzner et al. (997, 999) and Frey and McNeil (2002) postulated four axioms that should e met y a reasonale ( coherent ) measure of the whole risk capital WRC which after sutraction of overall expected losses yields the required regulatory capital: (Ax) WRC(L + L 2) WRC(L ) + WRC(L 2) (suadditivity), (Ax2) L L 2 WRC(L ) WRC(L 2) (monotonicity),

5 (Ax3) WRC( λ L) =λ WRC(L) for all λ 0 (positive homogeneity), (Ax4) WRC(L + L) = WRC(L ) + L for all certain losses L (translation-invariance). While (Ax) has already een explained, without (Ax2) ceteris parius higher losses with proaility one would possily result in lower requirements of WRC. (Ax3) just implies that the summation of an aritrary numer of perfectly correlated credits should lead to a corresponding increase in the whole risk capital, and according to (Ax4) certain losses have to e accounted for completely y additional capital requirements regardless of a ank s other loss risks. Additionally, (Ax4) implies the natural identity WRC(L WRC(L)) = 0 : The supply of the adequate amount WRC(L) of capital according to a loss distriution L reduces additional requirements to zero. Oviously, axioms (Ax) to (Ax4) sound quite reasonale. Moreover, as we want to show in the next section, as long as the whole risk capital satisfies these axioms, the optimal structure of a ank s credits may e independent of certain influence factors like the costs of regulatory capital and a ank s expected ankruptcy costs, thus simplifying a ank s portfolio decision to a great extent. 2. The setting To e more precise, consider a risk-neutral ank in a one-period-framework with uncertain cash flows c at time. Define B as the amount of money lent to orrower =,, at a contractual interest rate of r (B) and B = (B,..., B ˆ ) as the vector of all credits granted y the ank at time t = 0 to orrowers. Thus, B (+ r ) characterizes the amount of money orrower has to (B) return at t =. Certainly, some orrowers will not e ale to repay the whole amount B (+ r ). We use L(B) to descrie the ank s uncertain overall loss from such prolems. (B) 2

6 Moreover, the ank may lend or orrow on the interanking money market at a given interest rate r (f) for lending and orrowing with a total investment of A (A > 0: lending, A < 0: orrowing). We assume this lending and orrowing to e due just a short time efore credits and deposits are so that we can consider lending and orrowing on the interanking market to e risk-free. It seems plausile to assume that a ank s deposit rates are lower than r (f). Each ank will therefore choose the maximum possile amount D (max) of deposits, as choices D < D (max) would forgo riskless gains otherwise possile y investing additional deposits D (max) D at r (f) on the interanking market. We denote with E(r ) the corresponding expected (average) deposit rate to e paid (max) (D ) d y the ank under consideration. Under the plausile premise of inding anking capital regulation, the ank s required whole risk capital consists of provisions for expected losses E(L(B)) and the ank s (additional) regulatory capital RC(B). While we assume that the ank may acquire capital to account for ex- (WRC ) pected losses at a given interest rate r, additional capital requirements RC(B) may lead to (WRC 2 ) (RC) increasing (marginal) costs of capital r = :r. We thus have r (RC) (RC(B)) with (RC) (WRC ) r / RC 0 while r = const. Finally, we have to account for the ank s udget constraint B B + A = ˆ Now define (max) D + E(L(B)) + RC(B). E(c (B)) = (f ) (B) (max) (WRC ) (max) (D ) d = : = A ( + r ) + B ( + r ) E(L(B)) (2 + E(r )) D ( + E(r )) = B [r r ] E(L(B)) ( + E(r ) r ) D (E(r ) r ) (B) (f ) (max) (WRC ) (f ) (max) (D ) (f ) d () as the ank s expected cash flow efore sutracting additional expected costs of capital holding RC(B). A perfect ank regulation might accomplish that the proaility of default and thus 3

7 expected ankruptcy costs are (almost) independent of a ank s ehavior y requiring a certain regulatory capital RC. However, for real-life situations it sounds more plausile to assume that situations with ceteris parius higher requirements of RC coincide with higher anks default proailities and thus higher expected ankruptcy costs C(RC) for the ank under consideration: C(RC)/ RC 0. Summing up, we have (2) (RC) E(c (B)) = E(c (B)) RC(B) [ + E(r (RC(B)))] C(RC(B)) max.! B as the ank s ojective function. Ceteris parius, the ank will prefer alternatives with higher values for E(c ) and lower values for RC. 3. Implications The axioms (Ax) to (Ax4) for a ank s whole risk capital imply certain properties for its corresponding regulatory capital that serves as a defense against unexpected losses: Lemma. Let (Ax) to (Ax4) e true, then the regulatory capital RC = WRC EL ( ) satisfies (Ax), (Ax3) and the property (P4) RC( L + L ) = RC( L ) for all certain losses L. Proof. See the Appendix. RC thus possesses all relevant features that are known from the standard deviation as the most prominent measure of risk. Consider now a ank that is aiming at the maximization of the ojective function (2) and that is facing a legally defined RC that corresponds to WRC satisfying (Ax) to (Ax4). Let B (sum) e defined as = B, i.e. as the overall outstanding risky credits so that we may define y = B /B (sum) as the fractions of uncertain credits in the ank s overall risky credit suportfolio. Then, the Lemma enales us to derive the following 4

8 Proposition. ) Assume that an inner solution for the optimization of B (sum) does exist. Then optimal fractions y = B / B do not depend on * * ( sum)* (RC) E( r ( RC )), D (max), and r (D). Moreover, they are the same for all monotone increasing functions C(RC). Changes of the regulatory ank capital from RC(B) to f(rc(b)) with f(rc(b))/ RC(B) 0 do not affect the optimal credit portfolio structure, either. 2) Sufficient conditions for the existence of an inner solution for B (sum) to the ank s optimization prolem are expected ankruptcy costs C and/or expected costs ( ) ( RC ) Er of regulatory capital eing strictly convex in B (sum) given optimal credit portfolio structures. Proof. See the Appendix. According to our Proposition, as a consequence of coherent whole risk capital, the structure of a ank s optimal credit portfolio (if existing) does not depend on the general level of regulatory capital requirements, i.e. any tightening of capital requirements RC to f(rc) with f/ RC 0 will not affect the structure of a ank s risky suportfolio, ut only its amount. Moreover, a ank has not to other aout the precise functional relationship etween regulatory capital and expected ankruptcy costs as long as these relationships can e descried y aritrary monotone increasing functions. Further, it does not matter, either, how the expected costs of regulatory capital rise with a ank s additional demand for RC. Finally, the structure of the credit portfolio does not depend on the amount of deposits that is chosen y the ank nor on the corresponding deposit rate. The rationale for our Proposition is that, for given credit portfolio structure, variations of the amount of a ank s riskless lending or orrowing imply cominations of E(c ) and RC that all lie on a straight line with a fixed starting point ( E(c ) = (max) (f ) (max) (D ) d D (r E(r )); RC = 0) eing independent of the credit portfolio structure. All achievale lines thus differ only y their slopes and can therefore e unamiguously compared as long as a ank prefers ceteris parius 5

9 higher values for E(c ) and lower values for RC. The situation is analogously to that underlying the famous two-fund separation theorem of Toin (958) for mean-variance investors. 4. Discussion The separation result of the last section simplifies the ank s decision prolem tremendously and may serve as a further argument for coherent risk capital definitions. Certainly, our results are the consequences of some simplifying assumptions. In particular, we assume credits to e perfectly divisile and the ank to act as a pure price-taker at least on the credit market. However, since any single credit contriutes only to a minor degree to a ank s overall portfolio, the assumption of divisiility might e a satisfactory approximation of the real-life situation. Moreover, intense interanking competition will prevent anks from acting in another way than as price-takers. More importantly, we assume a possiility for riskless lending and orrowing on the interanking market, although we may take positive default proailities of a ank into account. We therefore must propose that lending and orrowing on the interanking market is due, efore credits and deposits are payale. Since interanking money market transactions are indeed often very short-term oriented, our setting seems to e reasonale. In addition, according to part 2) of the Proposition, even for C(RC) 0, i.e. (in particular) for situations without any default prolems for regulated anks so that interanking lending and orrowing are naturally riskless, an inner solution for the ank s optimization prolem may exist and thus the separation result remains in effect. Moreover, part ) of the Proposition will also hold true for situations with no riskless orrowing opportunity at all, as long as there is a possiility for riskless lending (y uying government notes and onds) and the ank management is sure to make use of this option. Summarizing, we deem it interesting to further analyze the portfolio selection consequences of coherent risk capital requirements in more detail. 6

10 AppendixFormelaschnitt (nächster) Proof of the Lemma: (Ax): WRC(L + L 2) WRC(L ) + WRC(L 2) RC(L + L 2) + E(L + L 2) RC(L ) + E(L ) + RC(L 2) + E(L 2). RC(L + L ) RC(L ) + RC(L ). 2 2 (Ax3): WRC( λ L) =λ WRC(L) RC( λ L ) + E( λ L ) =λ RC(L ) +λ E(L ) RC( λ L ) =λ RC(L ). (P4): WRC(L + L) = WRC(L ) + L RC(L + L) + E(L + L) = RC(L ) + E(L ) + L RC(L + L) = RC(L ). Proof of the Proposition: ) Define : = L /[B (+ r )] as the uncertain loss rate of credit. Then we have: (B) ˆ ˆ ˆ (B) (sum) (B) = = = L(B) = L = B (+ r ) = B y (+ r ). (A.) With this we get: (max) (B) (f) (WRC ) = + d (f) = E(c (B)) B (r r ) E(L(B)) ( E(r ) r ) D (E(r ) r ) = B [r r E( ) ( + r ) ( + E(r ) r )] D (E(r ) r ). = (B) (f ) (B) (max) (WRC ) (f ) (max) (D ) (f ) d (A.2) 7

11 For given deposits, a higher value of E(c (B)) corresponds with a higher value of (B) (f ) (B) (WRC ) µ (B): = B (f ) [r r E( )( + r )( + E(r ) r )]. (A.3) = For a given expectation value µ : = µ (B), the credit portfolios with a minimum of regulatory capital can e determined as follows: RC[B,..., B ] min.! (A.4) ( µ ) ( µ ) ˆ ( µ ) ( µ ) B,..., B (WRC ) s.t. µ = B ( µ ) (B) (f ) (B) (f ) [r r E( ) (+ r ) (+ E(r ) r )]. (A.5) = With λ µ as a Lagrangian multiplier, the necessary conditions are RC B ( µ ) =λ [r r E( ) ( + r ) ( + E(r ) r )] ( =,..., ) ˆ. (A.6) µ (B) (f ) (B) (WRC ) (f ) Since (from axiom (Ax3)) RC[c B,..., c B ] = c RC[B,..., B ] for all c 0, RC is ( µ ) ( µ ) ( µ ) ( µ ) ˆ ˆ homogeneous of degree one so that we get from Euler s Homogeneous Function Theorem ( µ ) ( µ ) ( µ ) RC ˆ ( µ ) = B RC B,..., B = B. (A.7) Under consideration of (A.5) and (A.6), the result (A.7) implies RC B,..., B =λ µ ( ) ( ) µ µ ˆ µ (A.8) Since (sum) µ ) (B) (f ) (B) (WRC ) (f ) = µ = B y [r r E( ) (+ r ) (+ E(r ) r )], (A.9) a variation of µ can e reached y a variation of B (sum) which in turn does not affect 8

12 λ = µ = Axiom (Ax3) (sum) ( µ ) y = µ ) (B) (f ) (B) + + (WRC ) (f ) = = RC B B y [r r E( ) ( r ) ( E(r ) r )] ( µ ) RC y =. y [r r E( ) (+ r ) (+ E(r ) r )] ( µ ) (B) (f ) (B) (WRC ) (f ) (A.0) As a consequence, for given credit portfolio structure (y,, y ) variations of B (sum) imply (µ, RC)-cominations that are all located on the same straight line. The optimal credit portfolio structure thus does not depend on the given values of µ and B (sum). Since the set of risky credit portfolios is convex in the µrc space due to the Lemma, the second-order conditions for a minimum regarding the prolem (A.4) are satisfied. A variation of D and r (D) does not influence the fact that the ank (following optimization prolem (2)) will prefer alternatives with higher values for µ (B) and lower values for RC. Thus, the proof aove is independent of the concrete specification of D and r (D). The same argument can e used to justify the last two sentences of part ) of the Proposition. 2) The ojective function (RC) E(c (B)) = E(c (B)) RC(B) [ + E(r (RC(B)))] C(RC(B)) (sum) ( (max) µ ) (B) (f ) (B) (WRC ) (f ) (max) (D ) (f ) d = = B y [r r E( ) (+ r ) (+ E(r ) r )] D (E(r ) r ) B RC y + E r B (sum) ( µ ) (RC) (su m) ( µ ) RC y = = (sum) ( µ ) C B RC y = (A.) 9

13 of the maximization prolem (2) implies an inner solution for B (sum), if C and/or (RC) E(r ) are strictly convex functions. References Artzner, Philippe, Freddy Delaen, Jean-Marc Eer and David Heath. (997). Thinking Coherently. Risk 0 (), Artzner, Philippe, Freddy Delaen, Jean-Marc Eer and David Heath. (999). Coherent Measures of Risk. Mathematical Finance 9 (3), Basel Committee on Banking Supervision. (2003). The Joint Forum: Trends in Risk Integration and Aggregation. (Download on 03/9/2006). Frey, Rüdiger and Alexander J. McNeil. (2002). VaR and Expected Shortfall in Portfolios of Dependent Credit Risks: Conceptual and Practical Insights. Journal of Banking and Finance 26, Kupiec, Paul. (2003). Understanding the Expected Loss Deate. Risk (), Toin, James (958). Liquidity Preference as Behavior Towards Risk. Review of Economic Studies 25,