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Characterise the Gaussian copula and demonstrate its use in deriving the chance that a pair of assets both default.

FRM® Part II · Market Risk Measurement and Management · Financial Correlation Modeling: Bottom-Up Approaches · FRM2.MR.10.b

The idea

The Gaussian copula links the marginals with a bivariate normal distribution whose correlation is ρ. Each name's cumulative default probability Q by time t is mapped to a normal threshold, x = N⁻¹(Q). A name defaults by t when its standard normal variable falls below that threshold. The thresholds for A and B are xA and xB, found from QA and QB in the same way. The chance that both A and B default is the bivariate normal cumulative at those thresholds, PAB = M(xA, xB, ρ). Setting ρ = 0 gives QA × QB, and ρ = 1 gives the smaller of QA and QB. The pairwise default correlation divides the covariance of the defaults by the product of their standard deviations, Corr = (PAB − PA × PB) ÷ (sA × sB). Each name's default indicator has a standard deviation given by sA = √(PA − PA²), and sB likewise. A common mistake is to treat ρ, the correlation of the latent normals, as the default correlation. For small default probabilities the default correlation is much lower. The conditional probability of B given A is PAB ÷ PA.

Formulas

Gaussian copula default threshold

x = N⁻¹(Q)

N⁻¹
the inverse standard normal cumulative distribution function
Q
the name's cumulative default probability to the horizon

Apply it to the default probability, not the survival probability, so the threshold is negative for small Q. With a constant hazard rate, Q is 1 − e^(−λt), not λt.

Joint default probability under a Gaussian copula

PAB = M(xA, xB, ρ)

M
the bivariate standard normal cumulative distribution function
xA
A's default threshold, N⁻¹(QA)
xB
B's default threshold, N⁻¹(QB)
ρ
the copula correlation between the latent normals

At ρ = 0 it equals QA × QB and at ρ = 1 the smaller of the two. It is a cumulative probability, not a density.

Correlation of two default indicators

Corr = (PAB − PA × PB) ÷ √(PA × (1 − PA) × PB × (1 − PB))

PAB
the probability that both names default
PA
the default probability of A
PB
the default probability of B

Subtract PA × PB before dividing. The result is far below the copula's ρ when default probabilities are small, so the two must not be confused.

A practice question

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

Under a Gaussian copula, the probability that two names A and B both default by time t is obtained by evaluating:

  1. the product N(QA(t)) x N(QB(t)), whatever the value of the correlation rho.Why not: Multiplies the marginals, which is the independence result, and applies N() to a probability rather than inverting it.
  2. the univariate normal cumulative distribution at the value rho x (QA(t) + QB(t)).Why not: Invents a combination of probabilities and rho; the bivariate normal cumulative with thresholds is required.
  3. the bivariate normal density at N^-1(QA(t)) and N^-1(QB(t)).Why not: Uses the density rather than the cumulative distribution; a default probability is an area, not a height.
  4. the bivariate normal cumulative at N^-1(QA(t)) and N^-1(QB(t)), with correlation rho. - correct

Default occurs when each latent normal variable falls below its threshold N^-1(Q(t)). The joint probability is the bivariate normal cumulative M(xA, xB; rho), which reduces to QA x QB only when rho is 0.

A harder one

Two names have default probabilities of 2% and 5%, and a model gives a probability that both default of 0.40%. The correlation of their default indicators is closest to:

  1. 0.003.Why not: Computes the covariance (0.004 - 0.001) and omits the division by the product of standard deviations.
  2. 0.098. - correct
  3. 0.131.Why not: Divides the joint probability 0.004 by the standard deviations without subtracting PA x PB.
  4. 0.200.Why not: Computes the conditional probability P(B defaults | A defaults) = 0.004 / 0.02, which is not a correlation.

Covariance = 0.0040 - 0.02 x 0.05 = 0.0030. The standard deviations multiply to sqrt(0.02 x 0.98 x 0.05 x 0.95) = 0.03051, so correlation = 0.0030 / 0.03051 = 0.098.

The hardest tier

Obligor X has a constant hazard rate of 5% a year and obligor Y has 2% a year. A bank links their default times with a Gaussian copula and wants the three-year joint default probability. Which pair of thresholds (X, Y) goes into the bivariate normal cumulative?

  1. (-1.04, -1.55).Why not: Uses lambda x t (0.15 and 0.06) as the cumulative default probability instead of 1 - exp(-lambda x t).
  2. (-1.66, -2.06).Why not: Uses the one-year default probabilities 1 - exp(-lambda) rather than the three-year horizon.
  3. (+1.08, +1.57).Why not: Wrong sign: applies the inverse normal to the survival probability exp(-lambda x t) instead of the default probability.
  4. (-1.08, -1.57). - correct

Q = 1 - exp(-lambda x 3): X gives 1 - exp(-0.15) = 0.1393 and N^-1 = -1.084; Y gives 1 - exp(-0.06) = 0.0582 and N^-1 = -1.570. Both thresholds are negative because they are lower-tail quantiles.

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