Calculating IRB formula for capital requirements in Basel II – The HangZhou Constant

By: Fred Vacelet, Eureka Financial Faculty

The Basel II text, in its IRB (Internal Rating Based) formula for capital requirements, ignores a few unpleasant properties of the calculations when PDs are low: capital requirements negative, division by 0, non-monotonicity, to name but a few.

Above a PD of 0.03% (the minimum for non-public entities), we note with delight that the curve is monotonically increasing in LGD, PD and M. This is what we intuitively expect. The potential for the capital requirement to be negative was dealt with swiftly: in that case, please set it to zero.

If we hold all the PD within a reasonable range, the curve bears all the poetry contained within the lips of Mona Lisa: mysterious enough to reveal no naughty thoughts, curvy enough, and smiling enough to tell that so much work and re-work had gone into its painting: for short, the start of a legend. Resemblances stop there, as the curve, unlike the painting, is monotonic, and shows little or nothing that was not expected. That is, for its visible part.

For sovereigns and supranational entities including the host of the Basel Committee, the PD is set to a Kelvin-type absolute zero (pure zero, not close to 0, not 10 to the power -99, but just zero as if impossible, unthinkable, no-go, taboo). Over-cynical minds will point out that it would have been politically incorrect if not blasphemy to suggest that a G10 government would ever fail. However, the fair laws of mathematics came to their rescue. The good old myth that countries and empires cannot fail makes the world believe that the probability of failure of the US is the Kelvin-type of absolute zero. Let us see a quick comparison with the Roman Empire, falling in A.D. 476 after some 1200 years of a remarkable success: assuming that the times of expected and actual death have coincided, the PD will have been 0.083%. For such a civilisation, it does not compare well with Enron, rated AAA, with a PD of 0.01%, who has had the sad privilege to go within a year from AAA to D with no return.

The curve, however, starts to be badly-behaved when we bring the PD into lower ranges, let alone when Excel and its rounding capabilities are used (you can find an Excel spreadsheet with the curve at: We first have the extreme discomfort of a zero denominator at the point where PD equals EXP((0.11852-SQRT(2/3))/0.05478): close to 0.00029%, corresponding to a failure for some 340,000 years of existence. Even before that infamous point in the curve, the curve is decreasing, meaning that the higher the PD, the lower the capital to hold.

There is a point below which the curve starts to exhibit its awful sides. We take the poetic licence to call the result the HangZhou constant, the point at which the first-order derivative turns to zero. The HangZhou constant, found to be approximately 0.0008745%, corresponds to every sovereign being bankrupt every 114,000 years. Whilst this comes short of eternity, this is more than enough to satisfy the ambitions of a few visionary members of the dictators’ club, whose main Nazi representative declared himself satisfied with 1000 years, and achieved a mere but bloody and tragic 12 of them. This is also more than enough quarterly periods to accommodate the horizon of modern democracies.

With a PD below the HangZhou constant, better use another function. A linear function will be a good start, providing us with the pleasures of:

  • Simplicity (over-simplicity, you might add? A spline will satisfy more people, as it avoids breakdown in derivatives at the HangZhou point)
  • monotonicity
  • continuity with the high-PD curve
  • concavity (granted, of 0, but not negative)

Sovereign models must not assume a PD of zero, but some methodological and practical issues can be avoided by the use of sovereign PDs always higher than the HangZhou constant. Politically, some hurdles remain for the use of the HangZhou constant: banks not wanting to add to their capital requirements, sovereign borrowers wanting to issue more and more debt, and few if any people happy to have yet another change in regulatory requirements.

If you want to learn more about Basel II / III and risk management and stress testing woofer in-company sessions on the following topics: Basel III & New Advances in Regulation and Enterprise-wide Risk Management & Stress Testing conducted by Fred Vacelet as well as many others. Check our public course calendar or contact us to discuss your requirements.


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