Abstract
Capital allocation rules are derived that maximize leverage while maintaining a target solvency rate for credit portfolios where risk is driven by a single common factor and idiosyncratic risk is fully diversified. Equilibrium conditions ensure that capital allocations depend on interest earnings as well as credits’ probability of default, endogenous loss given default, and asset correlation. Capitalization rates exceed those estimated using Gaussian credit loss models. Results demonstrate that credit risk is undercapitalized by the Basel II AIRB approach in part because of ambiguities regarding the definition of loss given default. An alternative proposed capital rule removes this bias.
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Notes
The capital allocation issues discussed herein apply to non-bank firms as well.
See, for example, the Basel Committee on Banking Supervision (1999).
The constraint sets a minimum bank solvency rate (1 minus the bank’s expected default rate).
In the Basel AIRB application of the GCLM, a bank must have loan loss reserves equal to EL or adjust its capital base. Reserves are a dynamic feature not incorporated into the static analysis in this paper, but corrections are made to account the absence reserve accounts.
See, Summary Findings of the Fourth Quantitative Impact Study (2006) (Office of the Comptroller of the Currency, Board of Governors of the Federal Reserve System, Federal Deposit Insurance Corporation, Office of Thrift Supervision 2006) and BCBS (2006a) for the results of the fifth quantitative impact study.
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.
See Kupiec (2004a) for pricing when the funding debt matures before the investment.
In the single asset case, when the probability of default on the purchased bond is less than or equal to (1 − α), the bond can be financed 100% with bank debt (Par F (α) = Par P ) without violating the solvency constraint.
Independence in this non-Gaussian setting requires that an observation of the return to bond j be uninformative regarding the conditional distribution function for bond i, \( \Pr {\left( {\left. {\widetilde{M}_{{it}} } \right|z_{M} } \right)} < a = \Pr {\left( {{\left( {\left. {\widetilde{M}_{{it}} } \right|z_{M} } \right)} < a{\text{given}}{\text{that}}\widetilde{M}_{{jt}} = M_{{jt}} } \right)},\forall a,i \ne j \). This condition is satisfied under the single common factor model assumption.
In the remainder of the discussion, consistent with Basle II capital rules, the horizon is assumed to be 1 year.
Some industry credit risk models include a stochastic default barrier such as in the Black and Cox (1976) model to increase the LGD relative to a basic BSM model and thereby improve correspondence with observed market data.
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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 for the comments of Rosalind Bennett, Charles Calomiris, Mark Fisher, Mark Flannery, George French, Charles Goodhart, Michael Gordy, Robert Jarrow, Wenying Jiangli, David Jones, Dan Nuxoll, Stuart Turnbull, Haluk Unal and Frank Zhang on an earlier draft of this paper. Email: pkupiec@fdic.gov
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Kupiec, P.H. Capital Allocation for Portfolio Credit Risk. J Finan Serv Res 32, 103–122 (2007). https://doi.org/10.1007/s10693-007-0013-4
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DOI: https://doi.org/10.1007/s10693-007-0013-4