Prioritize Risk Bucket Definition and Validation, Not PDFs

Eric Falkenstein

Moody’s Risk Management Services

Jan. 21, 2000

Basle reforms, RAROC pricing and capital arbitrage are all predicated on computations of economic risk capital, the amount of capital consistent with a target debt rating. In support of this objective the presentation of a credit probability distribution function (PDF) is now almost a cliché, and like a cliché it contains both wisdom and oversimplification. In my prior position as head of a capital allocations group, I found that workable solutions were invariably much less elegant than the PDFs would suggest. I presumed the discrepancy was due to my parochial view from the trenches, and that the larger, more sophisticated banks more closely approximated the ‘best practices’ one learns about at conferences and trade magazines. Subsequent experience has taught me that even the most advanced investment banks are as close to the theoretical ideal as Russia ever came to communism.

Economic capital allocation has a spurious simplicity because it is often presented in a language of which educated risk managers and academics knows each of the words. If all capital allocation discussions were forced to include a real numerical capital estimate (e.g., a retail banking branch should receive 75 basis points of capital for each dollar of deposits), it would be more difficult to be deluded into believing that an understanding of portfolio theory and the Merton model of equity are sufficient to get you 90% of the answer. The focus of capital allocation needs to move from PDFs, where the emphasis is on correlations, to product and exposure bucketing, where the emphasis is on validating loss or revenue variability assumptions.

The most common theoretical approach to internal capital allocation is best described as a credit Value at Risk. Capital estimation is analogous to value at risk, whereby the estimated probability of an adverse movement in asset values is equivalent to some acceptable insolvency rate. In this view the portfolio PDF is used to estimate an extremum used for determining required capital. The Basle Committee sponsored a study that puts this view succinctly: "In practice, the target insolvency rate is often chosen to be consistent with the bank’s desired credit rating … if the desired credit rating is Aa, the target insolvency rate might equal the historical one-year default rate for Aa-rated corporate bonds (about 3 basis points)." Figure 1 presents the ubiquitous graph:

Figure 1


The credit Value-at-Risk approach can be defined as an algorithm that statistically models future paths of a portfolio value or cash flows using assumptions about expected defaults, transitions, loss given defaults, correlations, and perhaps rating grade transitions credit spreads. When combined with regular value at risk, and the effect of market value on the exposure of unused commitments and derivatives, this approach becomes a complicated but tractable problem of portfolio statistics, derivative pricing formulas, yield curve models, etc.

This approach underlies many improvements to the current Basle regulatory standards. Such improvements include differentiating between low and high risk commercial loans, explicitly modeling the asymmetry of the loss profile, encouraging empirical validation, and the recognition of correlations.

Yet while we should welcome the improvements mentioned above, practitioners should see the credit VaR algorithm as a heuristic as opposed to an off-the-shelf formula. After a certain point parameter uncertainty implies that the ‘model’ is not analogous to any equations found in freshman physics texts. The difficulties in addressing the following issues cause me to think that this will always be the case:

Unresolved Issues

1) Required capital is misleading. What is really required is a cushion for an adverse event that includes book capital, cash flow, and access to outside funding. A company with little book capital after a large loss but significant franchise value is statistically more likely to be re-infused with cash, protecting creditors. Rating agencies, the Merton model of bankruptcy, and investors pay prominent attention to this part of the cushion, and so should any proposed solution. The profit cushion is also important, as many credit card operations in the U.S. suffered significantly higher losses than anticipated in 1996, yet still made money due to the large margins in that business. The profit cushion is rarely referenced in most credit VaR literature.

2) Underestimates the problem of correlations between lines of business. Frequently an algorithm is applied to a single portfolio, usually large corporate exposures, without regard to this portfolio’s correlation with other activities within a Large Complex Banking Operation offer such diverse lines of business as deposits, credit cards, home equity, auto lease, asset management, etc. A 99.97% extremum for a single portfolio to target consistency with a target Aa debt rating implicitly assumes a perfect correlation with other bank activities.

3) Ignores the importance of correlations with franchise value. Franchise value represents residual cash flows from assets not yet on the books, yet cash flows one can reasonable expected due to the history of the firm, its niche or brand value. If a firm’s portfolio were highly correlated with its franchise value (an extreme example would be stand-alone Special Purpose Vehicles), more book capital is required per amount of cushion.

4) Ignores the importance of liquidity. A firm with a highly volatile portfolio may be able to weather bad times if it can hypothecate various assets on its balance sheet (e.g., Citicorp), while other firms without this ability (e.g., LTCM) would not.

  1. Ignores the role of incentive conflicts in determining optimal capital levels. Theory relating to capital structure focuses upon capital as a bonding device that mitigates problems created by asymmetric information between the borrower and lender (see Hart’s Firms, Contracts and Financial Structure), they do not emphasize volatility of the underlying asset as the fundamental driver of capital levels. Instead, this field emphasizes the proportional level of equity in minimizing monitoring costs, reconciling differences of interests between debt holders, equity holders, and management, and minimizing adverse selection. These theories are not very useful for generating concrete implications, yet they can explain why often capital appears less a function of volatility than what the ‘capital as a cushion for unexpected loss’ approach suggests.

Lessons from the CDO market

Figure 2 shows the actual level of subordination to Baa as a function of expected loss for Collateralized Debt Obligations (CDOs) rated by Moody’s in 1996-98. The horizontal axis represents the lifetime losses assumed for the collateral given their weighted average agency rating and maturity, while the vertical axis is the size of subordination in the form of notes with ratings less than Baa (including unrated tranches). The vertical axis is analogous to the cushion provided by book capital to absorb an adverse event and yet provide consistency with the target default rate for the more senior notes. For all these CDOs the correlation was assumed to be almost the same degree as the Moody’s corporate universe.

Figure 2

The uncertainty demonstrated above is helpful in putting precise formulas into their proper perspective. Clearly there is a relationship between an objective measure of risk and required book capital for these CDO’s, yet also there are significant deviations from any best fitting line. Higher risk implies higher capital, but the relationship is imprecise because the cushion for the senior notes includes other factors such as liquidity facilities, excess spread, and the management quality of the issuer. These are exactly the types of considerations that materially affect an appropriate assessment of required risk capital within a financial institution.


A 99.97% target driving capital requirement is inconsistent with natural variability of time series data and bank balance sheets. The main benefit of the extremum approach to measuring risk is capturing nonlinearities not seen in one standard deviation moves, not to pinpoint capital to the fourth digit. The nebulous complexities involved in capital allocation, in addition to standard uncertainty in correlation and loss parameters, imply that reasonable differences of opinion can lead to material differences in capital, and this uncertainty is amplified the further out on the tail of the distribution one goes. A point estimate based largely on time series data is highly dependent on a handful of business cycles and will have standard errors large enough to make anything after the second digit meaningless and unverifiable. We should settle on targeting 99% extremums for a portfolio regardless of target debt rating, recognizing that the most interesting and important calculation is figuring out how to use that number in deriving the marginal capital requirement for a large complex financial institution.

Using credit VaR as part of an argument in favor of specific risk bucketing is compelling and in some cases necessary, but more should be written and said about the mundane issues that facilitate practical development today. These include how to best risk bucket a bank portfolio by product type (e.g., for home equity loans use loan-to-value, bureau scores, and geography) and what sort of practical rule is applied to generate capital factors for these risk buckets (e.g., 2.5 times expected losses for the direct consumer products). Most of the difficulty in allocating capital is mapping of credit exposures by line of business into risk buckets with validated expected loss estimates, or for products that statistically never lose money (e.g., deposits, fiduciary assets) generating net revenue histograms.

This emphasis is in contrast to approaches that emphasize correlations or portfolio models that generate PDFs as the most important issue. Most of these more academic approaches assume the expected loss and its variance already exist, and then address the issue of portfolio implications, just as financial theorists assumed in the Capital Asset Pricing Model. Let us hope that we will take less time than the CAPM theorists to recognize that such an approach is seriously limited in its practical application.

The Basle proposal on a new capital adequacy framework appropriately refers to internal models as a potentially useful method for assigning capital in the future. Turning potential into reality is not crucially dependent on Markowitzian portfolio statistics, but instead the painstaking process of bucketing exposures and products into homogeneous groupings and then estimating and validating net revenue variability for these groupings. The supervisory focus of Basle’s New Capital Adequacy Framework proposal properly emphasized supervision. In this vein, and both regulators and risk managers should spend less time with PDFs and more time beta testing tentative algorithms on actual business lines.


Basle Committee on Banking Supervision. Credit Risk Modelling: Current Practices and Applications. April 1999.

Basle Committee on Banking Supervision. A New Capital Adequacy Framework. June 1999.

Hart, Oliver. Firms, Contracts, and Financial Structure. 1995. Oxford University Press.