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Outline the Gaussian copula model and the uses to which it is put.

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

The idea

A copula combines separate marginal distributions into one joint distribution while controlling the dependence. The Gaussian copula maps each variable, percentile by percentile, onto a standard normal variable. It then assumes the normal variables are jointly normal with a chosen correlation matrix. For example, a firm's time to default with cumulative probability Q(t) is mapped to the standard normal value N⁻¹(Q(t)). The marginals can have any shape, so the original variables need not be normal. The copula correlation is not the correlation of the original variables. Applications include modelling default correlation among borrowers, valuing CDOs and nth-to-default swaps, and credit portfolio models such as Vasicek and CreditMetrics. In a one-factor version, each normal variable is built as X = √ρ × M + √(1 − ρ) × Z. Here M is the common factor and Z is the idiosyncratic term, independent standard normal variables. The model's main criticism is weak dependence in the extreme tails and a static correlation that proved too low under stress.

Formulas

One-factor Gaussian copula

z = √ρ × F + √(1 − ρ) × ε

ρ
the copula correlation with the common factor
F
the common factor, standard normal
ε
the borrower's idiosyncratic term, standard normal and independent of F

z is compared with thresholds N⁻¹ of cumulative probabilities. The correlation applies to the normals, not to the original variables.

A practice question

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

What does a Gaussian copula do in a credit risk model?

  1. It estimates each borrower's marginal default probability from its equity price and credit spread history.Why not: A copula takes the marginal distributions as given; it does not estimate them.
  2. It revalues loans after a rating change by discounting at a new credit spread.Why not: That is the revaluation step of a model such as CreditMetrics; a copula only specifies dependence.
  3. It links marginal distributions into a joint one by mapping each to a standard normal with a chosen correlation. - correct
  4. It forces every variable in the model, including times to default, to follow a normal distribution.Why not: Only the transformed variables are normal; the original marginals can have any shape.

The Gaussian copula separates marginals from dependence: percentile-to-percentile mapping to standard normals, with correlation between the normals defining the joint behaviour.

A harder one

A firm's time to default has constant hazard rate 4% per year, so the probability it defaults before time t is 1 - exp(-0.04 t). In a Gaussian copula, a default at t = 5 years is mapped to which value of the standard normal variable?

  1. 0.91Why not: Maps the survival probability instead of the cumulative default probability, flipping the sign.
  2. -0.91 - correct
  3. -0.20Why not: Uses the integrated hazard (lambda x t) directly as the normal value, without taking the percentile.
  4. -1.75Why not: Treats the annual hazard rate of 4% as if it were the cumulative probability by year 5.

Cumulative probability = 1 - exp(-0.2) = 0.1813. Normal value = N^-1(0.1813) = -0.91.

The hardest tier

Which statement about the Gaussian copula model is correct?

  1. The correlation of the normals equals the correlation between the original default times.Why not: The mapping is nonlinear, so the copula correlation differs from the correlation of the original variables.
  2. It requires the original variables, such as times to default, to be normally distributed.Why not: Only the transformed variables are normal; times to default are positive and typically exponential-like.
  3. It captures strong dependence in extreme tails, so joint extreme defaults are modelled well.Why not: A Gaussian copula has weak tail dependence, which is a main criticism: joint extreme events are understated.
  4. It specifies only dependence, so any marginals can be used; its correlation applies to the normals, not the original variables. - correct

The copula and the marginals are specified separately. Because the transformation is nonlinear, correlations differ across the two scales, and the Gaussian form has no asymptotic tail dependence, which is its key weakness in a crisis.

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