Calculation of Higher Moments in CreditRisk+ with Applications

CreditRisk+ is an influential and widely implemented model of portfolio credit risk. As a close variant of models long used for insurance risk, it retains the analytical tractability for which the insurance models were designed. Value-at-risk can be obtained via a recurrence-rule algorithm, so Monte Carlo simulation can be avoided. Little recognized, however, is that the algorithm is fragile. Under empirically realistic conditions, numerical error can accumulate in the execution of the recurrence rule and produce wildly inaccurate results for value-at-risk. This paper provides new tools for users of CreditRisk+ based on the cumulant generating function (“cgf”) of the portfolio loss distribution. Direct solution for the moments of the loss distribution from the cgf is almost instantaneous and is computationally robust. Thus, the moments provide a convenient, quick and independent diagnostic on the implementation and execution of the standard solution algorithm. Better still, with the cgf in hand we have an alternative to the standard algorithm. I show how tail percentiles of the loss distribution can be calculated quickly and easily by saddlepoint approximation. On a large and varied sample of simulated test portfolios, I find a natural complementarity between the two algorithms: Saddlepoint approximation is accurate and robust in those situations for which the standard algorithm performs least well, and is less accurate in those situations for which the standard algorithm is fast and reliable. ∗ The views expressed herein are my own and do not necessarily reflect those of the Board of Governors or its staff. I thank Tom Wilde and Paul Embrechts for helpful guidance and Gary Anderson for generous assistance with Mathematica. Please address correspondence to the author at Division of Research and Statistics, Mail Stop 153, Federal Reserve Board, Washington, DC 20551, USA. Phone: (202)452-3705. Fax: (202)452-5295. Email: 〈[email protected]〉. The CreditRisk+ model of portfolio credit risk has drawn significant practitioner interest since its publication in 1997 by Credit Suisse Financial Products.1 It appears to be currently the most widely-implemented of the so-called two-state or default-mode models, in which loss is recognized only in the event of borrower default before a fixed horizon date (typically one year ahead). Loss due to change in market value is ignored in these models. In particular, changes in borrower credit quality short of default, such as might be connoted by changes in agency rating, have no impact. The trend among practitioners at larger institutions appears to be towards adoption of multi-state models in which loss is defined on a mark-to-market basis. Multi-state modeling is indeed essential for pricing and trading applications and for proper treatment of the effect of loan maturity on credit risk. Nonetheless, two-state models do have advantages in the context of a buyand-hold lending portfolio. The default-mode definition of loss is more consistent with traditional book-value accounting, upon which the legal and regulatory standards of bank solvency depend. Two-state models may also be easier to calibrate using data readily available within the bank. Moreover, lending contracts often embed options that may be difficult to value precisely, especially in the context of a long-term and multi-faceted banking relationship.2 Pricing rules in the current generation of mark-to-market credit value-at-risk models are relatively simple and thus may provide a spurious sense of precision for some portfolios. As a practical matter, unless the portfolio is of very high quality or of long average duration, default risk is the dominant component of credit risk, so a default-mode measure of value-at-risk may be an adequate characterization of credit risk for many purposes.3 As a close variant of models long used for insurance risk, CreditRisk+ retains the analytical tractability for which the insurance models were designed. It yields a closed form solution for the probability generating function (“pgf”) of the distribution of portfolio credit loss. From this pgf, the probabilities associated with each possible level of credit loss can be calculated successively via a recurrence relationship. Provided that one chooses a reasonably coarse discretization of exposure sizes and parsimonious set of risk factors, the calculations can be performed quickly and accurately. This paper provides new tools for users of CreditRisk+ based on the cumulant generating function of the portfolio loss distribution. I show that the CreditRisk+ cumulant generating function (“cgf”) can be calculated quite easily and poses no numerical challenges. From the cgf, one can directly obtain the moments of the loss distribution. Though solution of tail percentiles in CreditRisk+ can be fast, direct solution of the moments can be orders of magnitude faster. Whereas solution time for tail percentiles increases exponentially with the number of risk factors and as a polynomial of the discretization of exposure sizes, solution time for moment calculations is invariSee Credit Suisse Financial Products (1997) for a complete description and technical discussion. For example, loan covenants may appear to give the bank power to renegotiate from a position of strength if, say, borrower operating income deteriorates, yet are routinely waived in practice. Loans to highly-rated borrowers typically have very small risk of default within the model horizon, so most of the credit risk in long-maturity, high-grade loans is associated with downgrade risk rather than default risk. In this case, default-mode measures may be seriously misleading. Lending books at most commercial banks are dominated by short to medium-term lending to borrowers of moderate credit quality.