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Contrast how the Merton model, CreditRisk+, CreditMetrics and the Moody's-KMV model each approach credit risk modelling, noting similarities as well as differences.

FRM® Part II · Credit Risk Measurement and Management · Introduction to Credit Risk Modeling and Assessment · FRM2.CR.5.f

The idea

The four models answer different questions. Merton treats equity as a call option on firm assets and defaults when assets fall below debt at maturity. The Moody's-KMV model builds on it. It infers asset value and volatility from equity and computes distance to default against a default point of short-term liabilities plus half of long-term liabilities. It then maps distance to default to an expected default frequency using a large empirical default database, rather than the normal curve. CreditMetrics is a portfolio model based on rating migration. It revalues each loan under every possible end-of-horizon rating, including default, using transition probabilities and asset or equity return correlations. CreditRisk+ is actuarial. It models default counts as Poisson with uncertain mean rates, says nothing about why firms default and needs no asset values. Merton, KMV and CreditMetrics are asset-value driven, while CreditRisk+ is not.

Formulas

Moody's-KMV distance to default

DD = (V − DP) ÷ (V × σ)

V
the market value of the firm's assets
DP
the default point, short-term liabilities plus half of long-term liabilities
σ
the volatility of the asset value

KMV maps the distance to an expected default frequency from its default database, not from the normal curve. Using total or only short-term liabilities as the default point is the trap.

Poisson probability of k defaults in a portfolio

P(k) = e^(−λ) × λ^(k) ÷ k!

λ
the expected number of defaults, the number of loans × the default probability
k
the number of defaults

CreditRisk+ treats default counts as Poisson. For at least k defaults, subtract the probabilities of fewer than k from one rather than reporting a single term.

A practice question

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

Which of the following credit models represents the number of defaults in a portfolio with a Poisson distribution?

  1. CreditRisk+ - correct
  2. CreditMetricsWhy not: CreditMetrics models rating migrations and values the loans under each rating, not Poisson default counts.
  3. Moody's-KMVWhy not: The KMV model maps a distance to default to an expected default frequency; it does not model default counts as Poisson.
  4. MertonWhy not: The Merton model derives default from asset value falling below debt, not from a count distribution.

CreditRisk+ is actuarial: the number of defaults is Poisson, with default rates that can themselves be random, and no assumption is made about the cause of default.

A harder one

A bank has a large portfolio of small retail loans for which it knows only default rates and their volatility. It has no equity prices and no ratings. Which framework is the most natural fit?

  1. Moody's-KMVWhy not: KMV requires equity prices to infer asset value and volatility, which are unavailable here.
  2. CreditMetricsWhy not: CreditMetrics needs ratings and a transition matrix, and asset or equity correlations, none of which are available.
  3. CreditRisk+ - correct
  4. MertonWhy not: Merton needs asset value and volatility, which cannot be obtained without equity prices.

CreditRisk+ needs only default rates, their volatility and exposures, and makes no assumption about the cause of default, so it suits retail portfolios.

The hardest tier

Under a CreditRisk+ style assumption, a portfolio of 200 loans, each with a one-year default probability of 1.5 percent, has Poisson-distributed defaults. The probability of at least two defaults in the year is closest to:

  1. 22.4%Why not: Computes the probability of exactly two defaults rather than at least two.
  2. 57.7%Why not: Computes the probability of more than two defaults, excluding exactly two.
  3. 95.0%Why not: Computes the probability of at least one default.
  4. 80.1% - correct

Mean defaults = 200 x 0.015 = 3. P(0) = e^-3 = 0.0498, P(1) = 3 e^-3 = 0.1494. P(at least 2) = 1 - 0.1991 = 80.1%.

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