Spell out a simulation-based way of arriving at credit losses for each tranche of a securitization.

FRM® Part II · Credit Risk Measurement and Management · Structured Credit Risk · FRM2.CR.22.h

The idea

Tranche credit losses are usually found by Monte Carlo simulation. In each trial, draw correlated default events for the collateral. A Gaussian copula is typical, in which every name has a latent variable built from a common market factor and an idiosyncratic shock. A name defaults when its latent variable falls below the threshold N⁻¹(PD). Apply the loss given default to the defaulted names to get the pool loss. Run that through the tranche attachment and detachment points, or the full waterfall, to get each tranche's loss, and repeat many thousands of times. Averaging over trials gives each tranche's expected loss, and percentiles give loss quantiles. Common errors are tranching the average pool loss instead of averaging the tranche losses, and using too few trials for senior tranches, whose losses occur in very few trials.

Formulas

Default threshold in a Gaussian copula

threshold = N⁻¹(PD)

N⁻¹: the inverse of the standard normal cumulative distribution
PD: the name's default probability over the horizon

A name defaults when its latent variable falls below this, so the threshold is negative for any PD below one half.

Default probability conditional on the market factor

PD(M) = N((N⁻¹(PD) − √ρ × M) ÷ √(1 − ρ))

N: the standard normal cumulative distribution
N⁻¹: its inverse
PD: the unconditional default probability
ρ: the asset correlation with the common factor
M: the draw of the common market factor

A bad market draw, M negative, raises the conditional PD. Multiply M by √ρ, not ρ, and divide by √(1 − ρ).

A practice question

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

Which sequence correctly describes the simulation approach to tranche credit losses?

Compute the average pool loss, allocate it to the tranches once, and multiply by the number of trials.Why not: Allocating the average loss once ignores the non-linearity of the tranche loss function; it should be applied trial by trial.
Simulate the tranche values directly from their market prices, then infer the collateral defaults.Why not: The simulation runs from collateral defaults to tranche losses, not the reverse.
Simulate each name's default independently, then scale tranche losses by the average correlation.Why not: Independent draws cannot represent default correlation, which drives tranche losses.
Simulate correlated defaults, compute the pool loss, allocate it to tranches, and average over many trials. - correct

Each trial draws correlated defaults, applies loss given default to get the pool loss and pushes it through the attachment and detachment points (or the full waterfall). The tranche expected loss is the average over trials.

A harder one

In a one-factor Gaussian copula with asset correlation rho = 0.30, a name has default probability 3 percent (threshold -1.881). Given a market factor draw of m = -2, the conditional default probability is N((threshold - sqrt(rho) x m) / sqrt(1 - rho)). Closest to this?

6.3 percentWhy not: Multiplies m by rho rather than by the square root of rho in the numerator.
0.02 percentWhy not: Adds sqrt(rho) x m instead of subtracting it, as if the market factor were favourable.
17.4 percent - correct
21.6 percentWhy not: Leaves out the division by sqrt(1 - rho), which overstates the conditional probability.

Numerator = -1.881 - 0.5477 x (-2) = -0.786; denominator = sqrt(0.7) = 0.8367; ratio = -0.939; N(-0.939) = 17.4 percent. A bad market draw raises the default probability from 3 to 17.4 percent.

The hardest tier

A simulation of a mezzanine tranche covering 5 percent to 15 percent of pool losses gives ten scenario pool losses of 0, 0, 1, 2, 3, 3, 5.5, 7, 12 and 17 percent. Closest to the estimated expected loss of the mezzanine tranche as a percentage of its notional?

0.5 percentWhy not: Applies the tranche loss function to the average pool loss of 5.05 percent, which barely exceeds the attachment point.
48.8 percentWhy not: Averages only over the four scenarios in which the tranche loses, ignoring the six scenarios with zero loss.
1.95 percentWhy not: Expresses the tranche loss as a percentage of the pool rather than of the tranche notional.
19.5 percent - correct

Tranche loss fractions are (L - 5)/10 capped between 0 and 1: 0, 0, 0, 0, 0, 0, 0.05, 0.20, 0.70, 1.00, which sum to 1.95. The mean over ten trials is 19.5 percent.

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