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Outline the difficulties involved in quantifying credit risk.

FRM® Part I · Valuation and Risk Models · Measuring Credit Risk · FRM1.VR.6.k

The idea

Quantifying credit risk is hard for several linked reasons. Defaults are rare, especially for high-quality borrowers, so estimates of default probabilities, correlations and recovery rates rest on few observations and are noisy. Capital models target a one-year, 99.9 percent loss, a one-in-a-thousand-year event. They cannot be properly backtested: with 25 years of history, a model that is badly wrong will usually show no exceedance, just as a correct one will. Loans are not traded, so market prices are missing, asset correlations are unobservable, and results are very sensitive to them. Recovery rates fall when default rates rise, so average LGD understates stress losses. Point-in-time PDs make capital procyclical. Large single-name concentrations break the granularity assumption of models such as Vasicek. Treat outputs as estimates with wide error bands, and test sensitivity to the inputs.

A practice question

From the question bank, with the reason each wrong option is wrong.

Why is it difficult to backtest a credit risk capital model built for a one-year horizon at a 99.9% confidence level?

  1. Regulators prohibit banks from comparing model output with the losses they actually realise.Why not: No such prohibition exists; the limitation is statistical.
  2. The model depends on daily market prices that are not available for loans.Why not: The difficulty is the rarity of tail events, not a lack of daily prices.
  3. An exceedance of the 99.9% level is expected only about once in a thousand years, so history cannot test the model. - correct
  4. Banks do not record credit losses, so no history exists against which to compare the model.Why not: Banks do record losses; the problem is that exceedances at 99.9% are too rare to test statistically.

At 99.9%, one exceedance per thousand years is expected, so a bank with 20 or 30 years of data cannot tell whether the model is right or wrong.

A harder one

A bank's credit capital model is set at a 99.9% one-year confidence level, but the true annual probability of a loss exceeding that level is actually 1%. Treating years as independent, the probability of observing no exceedances in 25 years of history is closest to:

  1. 22.2%Why not: Gives the probability of at least one exceedance, the complement of what is asked.
  2. 25%Why not: Multiplies 25 by 1%, which is the expected number of exceedances (0.25) not the probability of none.
  3. 77.8% - correct
  4. 97.5%Why not: Uses the claimed 0.1% exceedance probability instead of the true 1%.

P(no exceedance) = 0.99^25 = 77.8%. Even a model that understates tail risk by a factor of ten will usually show a clean record over 25 years, so the record cannot validate it.

The hardest tier

A bank estimates a PD of 1.0% for a low-default portfolio from sparse data. The true PD is 1.5%. Using a Vasicek model with copula correlation 0.12 and 99.9% confidence, and capital per unit of EAD x LGD equal to WCDR - PD, by what proportion of the correct capital is capital understated?

  1. 25.1%Why not: Compares the worst-case default rates without deducting expected loss from each, which understates the effect on capital.
  2. 23.9% - correct
  3. 31.4%Why not: Divides the gap by the understated capital instead of the correct capital.
  4. 33.3%Why not: Treats capital as proportional to PD, which is wrong: WCDR is a nonlinear function of PD and is net of EL.

Capital at PD 1.0% = 8.03%, at 1.5% = 10.56% of EAD x LGD. Understatement = (0.1056 - 0.0803) / 0.1056 = 23.9%. Noisy PD estimates feed straight into capital.

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