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Walk through how a synthetic CDO is priced, using both the spread payments method and a Gaussian copula model of default timing.

FRM® Part II · Credit Risk Measurement and Management · Credit Derivatives · FRM2.CR.13.f

The idea

A synthetic CDO is built from CDSs on a reference portfolio rather than owned bonds. Tranche investors act as protection sellers who receive a spread on the outstanding tranche principal and absorb losses once the attachment point is passed. In the spread payments method, the fair tranche spread makes the present value of expected premium payments equal to the present value of expected tranche losses. The premium payments fall as principal is written down. Expected losses require the joint distribution of default times, which the Gaussian copula supplies. Here M is a common factor, Z is an independent factor for each name, and ρ is the correlation. Each name's default time is mapped to a standard normal variable through x = √ρ × M + √(1 − ρ) × Z. A name defaults before time T if x is below the inverse standard normal of Q(T), the name's default probability by T. Higher correlation lowers equity tranche losses and raises senior tranche losses.

Formulas

Fair tranche spread by the spread payments method

s = PV(expected losses) ÷ PV(expected outstanding principal)

PV(expected losses)
the discounted expected tranche losses in each period
PV(expected outstanding principal)
the discounted expected tranche principal still outstanding at each payment date

Premiums are paid on the written-down principal, so charging them on the original notional understates the spread.

A practice question

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

In the Gaussian copula model for pricing a synthetic CDO, what role does the copula play?

  1. It supplies each name's marginal default probability by fitting a normal distribution to historical default rates.Why not: The marginal default probabilities come from CDS spreads (hazard rates); the copula only links them.
  2. It determines the recovery rate of each name as a function of the common factor in the model.Why not: In the standard model recovery rates are fixed inputs, not outputs of the copula.
  3. It joins each name's marginal default-time distribution into a joint distribution with a chosen correlation. - correct
  4. It removes dependence between defaults, so that tranche losses can be priced name by name.Why not: The copula introduces dependence; without it defaults would be independent and tranche pricing would be mispriced.

The marginals come from single-name CDS spreads; the Gaussian copula ties them together through a correlation parameter so that joint defaults, and thus tranche losses, can be simulated or integrated.

A harder one

In a one-factor Gaussian copula, a name has a five-year cumulative default probability of 5 percent, so the default threshold is N-inverse(0.05) = -1.6449. With a correlation of 0.30 and the common factor at M = -2, the conditional default probability is N(z) with z = (threshold - sqrt(rho) x M)/sqrt(1 - rho). The value of z is closest to:

  1. -0.66 - correct
  2. -1.25Why not: Uses rho instead of sqrt(rho) as the factor loading.
  3. -3.28Why not: Adds the common-factor term instead of subtracting it, which flips the effect of the adverse factor.
  4. -0.55Why not: Forgets to divide by sqrt(1 - rho), the scale of the idiosyncratic term.

z = (-1.6449 - 0.5477 x (-2))/0.8367 = (-1.6449 + 1.0954)/0.8367 = -0.66. A negative common factor raises the conditional default probability above 5 percent.

The hardest tier

A synthetic CDO references 100 names with USD 10 million notional each (USD 1,000 million) and a recovery rate of 40 percent on every name. The mezzanine tranche attaches at 3 percent and detaches at 7 percent. What is the minimum number of defaults needed to wipe out the mezzanine tranche completely?

  1. 7Why not: Ignores recovery and treats each default as losing the full 10 million.
  2. 5Why not: Counts only the defaults needed to exhaust the equity tranche (the attachment point), not the mezzanine detachment point.
  3. 12 - correct
  4. 18Why not: Uses the recovery of 4 million as the loss per default instead of the loss given default of 6 million.

Loss per default = 10 x 0.6 = 6 million. Detachment at 7% of 1,000 = 70 million. 70/6 = 11.67, so 12 defaults. (Attachment at 30 million is reached at the 5th default.)

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