FDIC Center for Financial Research Sanjiv R. Das Darrell Duffie Working Paper Nikunj Kapadia No. 2004-02 Capital Adequacy and Basel II Risk-Based Capital Standards, Deposit Insurance and Procyclicality Paul H. Kupiec Revised December 2004 Risk-Based Capital Standards, Deposit Insurance and Procyclicality Federal Deposit Insurance Corporation •C enter for Financial Researchh Capital Adequacy and Basel II by Paul H. Kupiec∗ September 2004 Revised December 2004 FDIC Center for Financial Research Working Paper No. 2004-02 ABSTRACT Using a one common factor Black-Scholes-Merton (BSM) equilibrium model of credit risk, this paper derives unbiased capital allocation rules for portfolios in which idiosyncratic risk is fully diversified. When these rules are compared with the Basel II Internal Ratings Based (IRB) minimum capital requirements for corporate exposures, the comparison shows that the Basel Advanced IRB (A-IRB) approach drastically undercapitalizes portfolio credit risk relative to the supervisory target of a 99.9 percent bank solvency rate. Estimates show that Basel regulations will allow fully compliant A-IRB banks to have default rates that exceed 5 percent. In contrast, the Foundation IRB minimum capital requirements allocate multiple times the capital necessary to achieve the supervisory objective. These results raise a number of important issues including the potential for increased systemic risk as regulatory capital rules promote banking sector consolidation and poor risk management standards in A-IRB banks. Key words: economic capital, credit risk, Basel II, internal models JEL Classification: G12, G20, G21, G28 CFR research programs: risk measurement, bank regulatory policy ∗ Associate Director, Division of Insurance and Research, Federal Deposit Insurance Corporation. The views expressed in this paper are those of the author and do not reflect the views of the FDIC. I am grateful to Mark Flannery, Dilip Madan, Haluk Unal, Jack Reidhill, Dan Nuxoll, Wenying Jiangli, and Andy Jobst for comments on an earlier draft of this paper. Email: [email protected]; phone 202-898-6768. Capital Adequacy and Basel II 1. INTRODUCTION Economic capital is the market value of equity in a bank’s capital structure. Equity capital functions as a buffer that protects all bank creditors from potential loss. Increases in equity capital raise the probability that a bank will fully perform on its contractual obligations. In practice, economic capital allocations often are estimated using value-at-risk (VaR) techniques that attempt to maximize bank leverage while ensuring that the potential default rate on a bank’s outstanding debt is below a maximum target rate selected by management.1 Regulatory capital requirements set legally binding minimum capitalization standards for banks. Supervisory measures of capital differ from theoretical measures of economic capital in that the former may include selected classes bank liabilities such as subordinated debt, trust preferred stock, loan loss reserve balances, and other so-called hybrid capital instruments in the bank’s loss absorbing buffer. Relative to economic capital, additional components are included in the regulatory measure because regulators and bank managers have different objective functions. A primary objective of bank regulatory policy is to preserve the integrity of banks’ core transactions and savings deposits in the event of insolvency while allowing losses to accrue to hybrid forms of bank capital. In contrast, managers must capitalize to ensure that a bank is a going-concern, and to remain a going concern, banks must perform fully on all of their liabilities. Under the June 2004 Basel II agreements, national supervisory authorities may choose among three alternative minimum regulatory capital frameworks for their internationally active banks. One approach, the standardized approach, sets minimum capital standards using a modified version of the 1988 Basel Capital Accord that links capital 1 Alternatively, the constraint can be described as a minimum bank solvency margin (1 minus the bank’s expected default rate). - 2 - requirements to external credit ratings. The remaining two approaches, the so-called Foundation (F-IRB) and Advanced (A-IRB) Internal Ratings Based approaches, use mathematical rules to assign minimum capital requirements according to an individual credit’s probability of default (PD), loss given default (LGD), maturity, and exposure at default. In the U.S., banking supervisors have determined that Basel II implementation will require so-called core banks to adopt the A-IRB approach, while other banks may petition supervisors for A-IRB capital treatment (so-called opt-in banks). The remaining banks (so- called general banks) will continue using the capital requirements specified in the 1988 Basel Capital Accord.2 The Basel II IRB framework has been distilled from a special class of VaR models, the so-called asymptotic single risk factor (ASRF) class, in part because these models generate portfolio-invariant capital requirements that can be used to calibrate IRB capital assignment rules.3 In designing its prudential standards, the Basel Committee on Banking Supervision (BCBS) has expressed an objective that includes setting minimum regulatory capital requirements at a level consistent with a minimum bank solvency margin of 99.9 percent over a one-year horizon.4 In practice, the BCBS arrived at the June 2004 IRB framework through a process in which successive IRB calibrations were modified in response to industry commentary and information gleaned from Quantitative Impact Studies (QIS). A series of three QIS studies required banks to estimate the effects of alternative IRB calibrations on their minimum regulatory capital requirements. Consecutive IRB formulations were modified toward a goal of achieving capital neutrality while creating incentives that encouraged banks to adopt the IRB approaches.5 While the QIS exercises 2 Core banks are defined as institutions with total banking (and thrift) assets of $250 billion or more or total on-balance-sheet foreign exposure of $10 billion or more. General banks likely will be subject to a modified version of the 1988 Basel Accord, but the potential modifications have yet to be publicly discussed. 3 See for example, Gordy (2003). 4 Basel Committee on Banking Supervision, 2004, paragraph 667. 5 The Basel Committee on Banking Supervision, October 2002, paragraphs 46-47 - 3 - have provided the BCBS with information on the minimum capital requirements that banks would face under Basel II’s alternative approaches given their existing portfolios, the QIS studies do not provide any evidence regarding the minimum solvency margins implicit in the alternative Basel II approaches. To date, no published study has analyzed the rigor of the prudential standards that are set by the June 2004 Basel II IRB model calibrations. This paper analyzes the minimum solvency standards associated with the Basel II IRB approaches using the Black-Scholes-Merton (BSM) equilibrium model of credit risk. Semi-closed form solutions for the capital allocation problem are derived using the physical and equivalent martingale return distributions for credit portfolios that satisfy the ASRF assumptions—default correlations are driven by a single common factor and idiosyncratic risk is fully-diversified. Since optimal capital allocation rules are derived from a full BSM portfolio model, the capital rules fully anticipate the correlations that arise among individual credit’s probability of default and loss given default. The solvency margins implicit in the June 2004 IRB specifications are estimated in a calibration exercise that compares IRB capital requirements to the unbiased capital allocations specified by the BSM ASRF model. The results of the calibration comparison have important regulatory and competitive implications. A comparison of the alternative capital allocation approaches suggests that, compared to the true capital needed to ensure a regulatory target solvency margin of 99.9 percent, the June 2004 A-IRB approach undercapitalizes banks by more than 80 percent assuming that all bank regulatory capital is Tier 1 (equity) capital. The A-IRB capital shortfall is shown to be consistent with bank default rates that are in excess 5 percent. Since Basel II allows banks to use subordinated debt and other hybrid forms of capital to satisfy minimum regulatory requirements, the potential default rates of fully compliant A-IRB banks could be in excess of 5 percent. If widely adopted, the regulatory standards set by the A-IRB approach will allow a system-wide bank insolvency rate similar in magnitude to the 6.4 percent failure rate experienced by insured U.S. savings and loan institutions during the height of the 1980s crisis.6 These analysis shows that June 2004 A-IRB approach is seriously flawed. The A- 6 “History of the Eighties—Lessons for the Future,” p. 168. - 4 - IRB represents a very poor standard for measuring the quality of bank risk measurement practices. The calibration analysis indicates that, unlike A-IRB banks, F-IRB banks will be over-capitalized relative to the 99.9 percent target solvency rate. For high quality (low default risk) portfolios, F-IRB capital requirements specify more than seven times the level of capital needed to achieve the regulatory target. F-IRB capital requirements for lower quality (high default risk) credits are overstated by about 170 percent. The overcapitalization owes in part to the F-IRB assumption that LGD is 45 percent, an assumption that overestimates the loss rates on the credits examined in this study. For short-maturity credits, F-IRB capital requirements provide capital relief relative the 8 percent required by the 1988 Basel Accord. For longer maturity credits, the 1988 Basel Accord assigns lower capital requirements. Overall, the analysis demonstrates that the F-IRB prudential standard is much more conservative than the standard that will apply to A-IRB banks. In designing Basel II, the BCBS calibrated the alternative approaches so that, for a wide range of credits, the A-IRB approach produces the lowest capital requirements. This calibration is designed to encourage banks to transition from the Standardized and F-IRB approaches to the A-IRB approach. Capital savings accorded under the A-IRB are intended to offset the costs associated with developing and operating A-IRB systems and to reflect efficiencies that are presumed to be generated by the A-IRB’s more efficient measurement of credit risk and assignment of minimum capital. While the BCBS have succeeded in creating incentives that encourage adoption of the IRB approaches, the large differences in the regulatory capital requirements specified by the alternative Basel II IRB approaches raise important prudential and structural issues. To the extent that banks enjoy safety-net engendered subsidies that are attenuated by minimum regulatory capital requirements, the IRB calibrations engender strong incentives that will encourage banking system assets to migrate toward A-IRB banks in order to maximize the value of the implicit safety net subsidy. Asset migration could be achieved through consolidation or through an increase in the number banks that are granted regulatory approval for the A-IRB approach. Regardless of how asset migration occurs, the capital relief granted under the A-IRB approach may raise long-term prudential concerns. If the - 5 - minimum leverage requirements of FDICIA’s prompt corrective action regulations are relaxed once Basel II is implemented, unless market discipline forces intervene and “require” banks to maintain additional capital buffers, regulations will allow A-IRB banks to operate with substantial reductions in regulatory capital.7 Should A-IRB capitalization levels approach Basel II regulatory minimums, the associated solvency margin estimates portend a material increase in bank failure rates to levels beyond those experienced in the 1980s banking crisis. An outline of this paper follows. Section 2 summarizes the general methodology for constructing unbiased economic capital allocations. Section 3 revisits unbiased credit risk capital allocation in the context of the Black-Scholes-Merton (BSM) model. Section 4 derives unbiased portfolio-invariant credit risk capital measures for an ASRF version of the BSM. Section 5 reviews the procedures for setting minimum capital requirements under the Basel IRB approaches. Section 6 discusses the calibration results and Section 7 concludes the paper. 2. UNBIASED BUFFER STOCK CAPITAL FOR CREDIT RISKS The intuition that underlies the construction of an unbiased economic capital allocation for a portfolio is transparent when considering portfolios composed of long positions in traditional financial assets such as simple loans, bonds, or equities because the value of the investment portfolio cannot become negative. For instruments on which losses have the potential to exceed their initial market value, as they can for example on short positions, futures, derivatives, or other structured products, then economic capital calculations must be modified from the techniques described subsequently because capital structure modifications alone may not be able to ensure that a bank is able to perform on its liabilities. In these circumstances, ensuring a minimum solvency margin may require 7 Unless prompt corrective action (PCA) minimum capital requirements are relaxed (12 U.S.C. Section 1831), PCA may become the binding regulatory capital constraint on A-IRB banks. - 6 - changing portfolio investment shares.8 For purposes of the analysis that follows, portfolio composition is restricted to include only long positions in fixed income claims that may generate losses that are bounded above by the initial market value of the credit. Defining an Appropriate Value-at-Risk (VaR) Measure Let T represent the capital allocation horizon of interest. The purchased asset A, has ~ an initial market valueA , a time T value of A , that has an associated cumulative 0 T ~ distribution function represented by Ψ(A ,A ), and a probability density function T T ~ (~ ) represented by ψ(A ,A ). Let Ψ−1 A ,1−α represent the inverse of the cumulative density T T T ~ ( ) [ ] function of A evaluated at 1−α. Define an α coverage VaR measure, VaRα,α∈ 0,1, T as, (~ ) VaR(α)= A −Ψ−1 A ,1−α (1) 0 T ( ) VaRα measures the loss that could be exceeded by at most (1−α) of all potential future ~ value realizations ofA . Note that expression (1) measures value-at-risk relative to the initial T market value of the asset. When credit risk losses are bounded above by the initial invested (~ ) amount, A , Ψ−1 A ,1−α is bounded below by 0. 0 T Unbiased Capital Allocation for Credit Risk Assume, hypothetically, that a bank follows a capital allocation rule that sets equity ( ) capital equal toVaR .999 . By definition, there is less than 0.1 percent probability the ( ) investment’s value will ever post a loss that exceeds the VaR .999 measure. The amount ( ) that must be borrowed to finance this investment asset under this VaR .999 capital allocation rule is A −VaR(.999). If the bank borrowsA −VaR(.999), it must promise to 0 0 pay back more than A −VaR(.999) if equilibrium interest rates and credit risk compensation 0 ( ) are positive. Because the VaR .999 capital allocation rule ignores the equilibrium returns 8 See Kupiec (2004) for a discussion of the capital allocation problem in these instances. - 7 - that are required by bank creditors, the probability that the bank will default on its funding ( ) debt under a VaR .999 capital allocation rule is greater than 0.1 percent if the bank’s debts ~ can only be satisfied by funds raised from selling the purchased asset forA at time T.9 T An unbiased economic capital allocation rule for 0.1 percent target default rate is: set equity capital equal to VaR(.999) plus the interest that will accrue on the bank’s borrowings. Alternatively, set the par (maturity) value of the funding debt equal to VaR(.999)and estimate its current equilibrium market value. The difference between the current market value of the purchased asset and the current market value of the associated funding debt is the economic capital needed to fund the investment and satisfy the solvency rate target. This capital allocation rule generalizes to the portfolio context. In order to estimate the equilibrium interest cost on funding debt, one must go beyond the tools of value-at-risk and employ formal asset pricing models or use empirical approximations to value bank’s funding debt. The following section modifies the Black- Scholes-Merton (BSM) model to price the bank’s funding debt issue. 3. Unbiased Buffer Stock Capital Allocation in a Black-Scholes-Merton Model If the risk-free term structure is flat and a firm issues only pure discount debt, and asset values follow geometric Brownian motion, under certain simplifying assumptions10, Black and Scholes (1973), and independently Merton (1974) (hereafter BSM) established that the market value of a firm's debt issue is equal to the risk free discounted value of the bond’s par value, less the market value of a Black-Scholes put option written on the value of the firm’s assets. The put option has a maturity identical to the debt issue maturity, and a strike price equal to the par value of the debt. More formally, if A represents the initial 0 9 A formal proof is given in Proposition 1 in the Appendix. 10 There are no taxes, transactions are costless, short sales are possible, trading takes place continuously, if borrowers and savers have access to the debt market on identical risk- adjusted terms, and investors in asset markets act as perfect competitors. - 8 - value of the firm’s assets, B the bond’s initial equilibrium market value, and Parthe bond’s 0 promised payment at maturity date M, BSM establish, B = Par e−rfM −Put(A ,Par,M,σ), (2) 0 0 ( ) where r represents the risk free rate and Put A ,Par,M,σ represents the value of a Black- f 0 Scholes put option on an asset with an initial value of A , a strike price of Par, 0 maturityM,and an instantaneous return volatility of σ. The default (put) option is a measure of the credit risk of the bond. Merton (1974), Black and Cox (1976), and others show that the model will generalize as to term structure assumptions, coupon payments, default barrier assumptions, and generalized volatility structures. The capital allocation discussion that follows uses the simplest formulation of the BSM model.11 Modifying the BSM Model for Credit Risk Capital Allocation In the original BSM model, the underlying assets exhibit market risk. To examine portfolio credit risk issues, it is necessary to modify the BSM model so that the underlying assets in the bank’s portfolio are themselves risky fixed income claims. Consider the case in which a bank’s only asset is a risky BSM discount debt issued by an unrelated counterparty. Assume that the bank will fund this bond with its own discount debt and equity issues. In this setting, the bank’s funding debt issue can be valued as a compound option. ~ Let A and Par represent, respectively, the time T value of the assets that support T P the discount debt investment and the par value of the bond. Let Par represent the par value F of the discount bond that is issued to fund the investment. For purposes of simplifying this discussion we restrict attention to the case where the maturity of the bank’s funding debt 11 That is, it assumed that the term structure is flat, asset volatility is constant, the underlying asset pays no dividend or convenience yield, and all debt securities are pure discount issues. - 9 -