Charta is on Google Play: practise on the go. Get the app
Charta

Explain how each of Vasicek's model, CreditRisk+ and CreditMetrics derives the probability distribution of default losses and models default correlation.

FRM® Part II · Credit Risk Measurement and Management · Credit Value at Risk · FRM2.CR.10.e

The idea

The three models build the default-loss distribution differently. Vasicek's model uses one normally distributed systematic factor with a common asset correlation. For a large homogeneous portfolio it gives a closed-form distribution of the default rate. CreditRisk+ is actuarial. The number of defaults is Poisson with stochastic mean default rates. Defaults are correlated because obligors share background factors that move their default intensities together. Exposures are banded and the loss distribution comes from an analytic recursion, with default the only event. CreditMetrics is structural and multi-state. Correlated asset returns, typically derived from equity data, drive rating migrations. Bonds are revalued at each outcome, and the distribution is usually found by Monte Carlo simulation.

A practice question

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

Which model treats the number of defaults as Poisson-distributed and derives the loss distribution with an analytic actuarial approach?

  1. CreditMetricsWhy not: A migration and revaluation model with simulated asset returns; it is not Poisson-based.
  2. CreditRisk+ - correct
  3. Vasicek's single-factor modelWhy not: Uses a normal latent factor and gives a closed-form default-rate distribution, not Poisson counts.
  4. The Merton structural modelWhy not: Derives default from asset value falling below debt and is not an actuarial frequency model.

CreditRisk+ is borrowed from insurance mathematics: Poisson default counts with stochastic mean rates, exposure bands and an analytic recursion for the loss distribution.

A harder one

A portfolio is expected to have 3 defaults a year and defaults follow a Poisson distribution with constant mean. The probability of 5 or more defaults in a year is closest to:

  1. 18.5% - correct
  2. 10.1%Why not: Gives the probability of exactly 5 defaults, not 5 or more.
  3. 8.4%Why not: Gives the probability of 6 or more defaults.
  4. 35.3%Why not: Gives the probability of 4 or more defaults.

P(k) = e^-3 x 3^k / k!. P(0..4) = 0.0498 + 0.1494 + 0.2240 + 0.2240 + 0.1680 = 0.8153. P(5 or more) = 1 - 0.8153 = 18.5%.

The hardest tier

In Vasicek's large-portfolio model, PD = 2% and asset correlation = 0.10. The default-rate distribution is P(DR < x) = N[(sqrt(1 - rho) x N^-1(x) - N^-1(PD)) / sqrt(rho)], with N^-1(0.02) = -2.054 and N^-1(0.05) = -1.645. The probability that the default rate exceeds 5% is closest to:

  1. 94.1%Why not: Reports P(DR < 5%) instead of the exceedance probability.
  2. 3.5%Why not: Uses 0.9 in place of sqrt(0.9) in the numerator of the normal argument.
  3. 2.0%Why not: Mistakes the unconditional PD for the exceedance probability of a 5% default rate.
  4. 5.9% - correct

Argument = (0.9487 x -1.645 + 2.054) / 0.3162 = 1.560. N(1.560) = 94.1%, so P(DR > 5%) = 5.9%.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Credit Value at Risk