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Account for how the distribution of credit losses is modelled.

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

The idea

Credit losses are not modelled as normal. The loss distribution is bounded at zero, with a high probability of small losses and a long right tail from clustered defaults, so the mean, EL, sits well below the tail quantiles that drive capital. It is built from three random pieces: default (binomial or Bernoulli, with PD), the loss rate, and exposure. The loss rate is often fitted with a beta distribution because it lives between 0 and 1. Correlation among obligors, mainly through common economic factors, fattens the tail, as does concentration in few large exposures. Because the combined distribution has no closed form, banks use Monte Carlo simulation or analytic approximations. They then read the required quantile off the simulated losses. The exam point: skewness makes normal-based multiples of UL understate capital.

Formulas

Variance of a beta-distributed loss rate

variance = m × (1 − m) ÷ (α + β + 1)

m
the mean loss rate, α ÷ (α + β)
α
the first beta parameter
β
the second beta parameter

Solve for α + β from the target variance, then α = m × (α + β). The beta keeps the loss rate between 0 and 1.

A practice question

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

Which description best fits the distribution of credit losses on a loan portfolio over one year?

  1. Skewed to the right with a long, fat tail, with expected loss well below the tail loss. - correct
  2. Symmetric and normal, with the expected loss at the centre of the tail.Why not: Assumes normality, which understates the probability of large losses.
  3. Uniform between zero and the total exposure.Why not: Ignores that small losses are far more likely than large ones.
  4. Concentrated at the expected loss, since defaults are independent.Why not: Ignores variability; diversification reduces but never eliminates it.

Many small losses and rare, clustered large ones produce a right-skewed, fat-tailed distribution, so tail-based capital exceeds what a normal fit suggests.

A harder one

A bank raises the correlation between obligors' defaults, leaving each PD, loss rate and exposure unchanged. What happens to the credit loss distribution?

  1. Expected loss rises along with the tail.Why not: EL is a sum of individual ELs and does not depend on correlation.
  2. Expected loss falls and the distribution narrows.Why not: Gets the effect of correlation backwards.
  3. Nothing changes, since PDs are unchanged.Why not: Ignores that default clustering widens the distribution without changing its mean.
  4. Expected loss is unchanged, but the tail fattens, so UL and the loss quantile rise. - correct

Correlation leaves the mean (EL) alone but makes joint defaults more likely, raising UL and the tail quantile.

The hardest tier

A loss rate is modelled with a beta distribution whose mean is alpha/(alpha + beta) and whose variance is m x (1 - m)/(alpha + beta + 1), where m is the mean. For a mean loss rate of 40 percent and a standard deviation of 20 percent, the parameters are:

  1. alpha = 2, beta = 3. - correct
  2. alpha = 2.4, beta = 3.6.Why not: Uses alpha + beta = m(1-m)/variance and forgets the +1 in the denominator.
  3. alpha = 3, beta = 2.Why not: Swaps the parameters, giving a mean of 60 percent.
  4. alpha = 4, beta = 6.Why not: Doubles both parameters, which keeps the 40 percent mean but gives a standard deviation of only about 15 percent.

Variance 0.04 = 0.24/(alpha+beta+1) gives alpha+beta = 5; alpha = 0.4 x 5 = 2, beta = 3.

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