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Outline what makes credit risk difficult to quantify.

FRM® Part II · Credit Risk Measurement and Management · Capital Structure in Banks · FRM2.CR.4.i

The idea

Credit risk is hard to quantify for several reasons. Defaults are rare and clustered, so there is little data, and a 99.9 percent one-year loss needs a thousand-year event to validate directly. Loans are mostly illiquid and not marked to market, giving no observable price history. The loss distribution is skewed and fat-tailed, so standard deviations alone mislead. Default correlations are small, unstable and hard to estimate, yet they drive the tail. PD and LGD tend to rise together in downturns, so averages understate losses. Parameters are themselves uncertain, which adds systematic risk that diversification cannot remove. The result is high model risk. The exam point: backtesting a high-confidence credit loss estimate is nearly impossible with realistic history.

Formulas

Expected loss rate when PD and LGD are correlated

E[PD × LGD] = E[PD] × E[LGD] + Cov(PD, LGD)

E
the expectation
PD
the probability of default
LGD
the loss rate
Cov(PD, LGD)
the covariance of PD and the loss rate

Positive covariance in downturns means multiplying average PD by average LGD understates expected loss.

A practice question

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

What is a principal reason credit risk is harder to quantify than market risk?

  1. Defaults are rare, so historical data are scarce and loans seldom have observable prices. - correct
  2. Credit exposures are always small, so they are measured with large errors.Why not: Exposure size is not the cause of measurement difficulty.
  3. Credit losses are normally distributed, so standard formulas fail.Why not: Credit losses are skewed and fat-tailed, not normal; the claim is also not why data are scarce.
  4. Default probabilities are always known exactly, leaving nothing to estimate.Why not: Reverses the problem: PDs are estimated with great uncertainty.

Few defaults, long horizons and illiquid, unpriced loans leave little data from which to estimate PD, LGD and correlation.

A harder one

A bank has 20 years of annual loss data and wants to validate a 99.9 percent annual loss quantile. If the model is exactly right, the probability that the sample shows no exceedance is closest to:

  1. 2.0 percent, so an exceedance is likely.Why not: Gives the probability of at least one exceedance rather than none.
  2. 0.1 percent, so a clean record would be very unlikely if the model were right.Why not: Uses the single-year exceedance probability and ignores compounding over 20 years.
  3. 99.9 percent, so the model is validated by a clean record.Why not: Uses the single-year figure instead of raising it to the power 20.
  4. 98.0 percent, so a clean record says almost nothing about the model. - correct

P(no exceedance in 20 years) = 0.999^20 = 0.9802. Because most sound and unsound models both show a clean record, validation is weak.

The hardest tier

A bank prices a USD 10 million exposure assuming independent PD and loss rate with means of 3 percent and 40 percent. In practice the correlation between them is 0.5, with standard deviations of 2 percent for PD and 10 percent for the loss rate. By how much does the independence assumption understate expected loss?

  1. USD 10,000. - correct
  2. USD 130,000.Why not: Reports the full correct EL rather than the amount of understatement.
  3. USD 100,000.Why not: Uses the correlation times the PD standard deviation, leaving out the loss-rate standard deviation.
  4. USD 20,000.Why not: Multiplies the standard deviations but ignores the correlation.

E[PD x LGD] = 0.03 x 0.40 + 0.5 x 0.02 x 0.10 = 0.013, so EL = 130,000 vs 120,000 under independence: gap 10,000.

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