Gaussian Copula Credit Model

A CDS spread tells you one company's own probability of default. A bank holding 100 corporate bonds needs more than 100 separate numbers — it needs to know how likely it is that many of them default together, since individually-fine bonds can still wipe out a portfolio if their defaults cluster in one bad year. The Gaussian copula model exists to answer that: given each bond's own default probability, how do you glue them into one joint probability of a bad outcome?

Picture 100 dominoes on a table, each wired to a shared string underneath. Pull the string a little and a few wobble; pull it hard and most fall together. The string is the economy — a single shared factor. Each domino's own stability is its individual default probability; how tightly it's tied to the string is its correlation to everyone else. The Gaussian copula writes down "how tightly tied to the string" precisely.

The mechanics

Each company $i$ gets a hidden "asset value" variable built from two pieces:

$$X_i = \rho \, M + \sqrt{1-\rho^2}\, Z_i .$$

$M$ is the single shared factor — the string, the same for every company. $Z_i$ is company $i$'s own private noise, independent of everyone else's. $\rho$ (between 0 and 1) is the correlation: how much of company $i$'s fate is tied to the shared string versus its own idiosyncratic luck. Company $i$ defaults if $X_i$ falls below a threshold $c_i$, chosen so that probability exactly matches the company's own known default probability, read off its CDS curve. No new information about why companies default is added — the model wires already-known individual probabilities together through one shared factor.

Worked example 1 — two bonds, one shared factor

Two bonds each have a 5% chance of defaulting over 5 years, and their asset values share correlation $\rho = 0.3$. If they were independent, the chance both default is just $0.05 \times 0.05 = 0.0025$, or 0.25%. But they aren't independent — both are exposed to the same $M$. Feeding $\rho = 0.3$ and the 5% marginal thresholds into the joint (bivariate) normal distribution pushes the joint default probability up to roughly 0.9% — more than three times the independent case. The gap between 0.25% and 0.9% is entirely the price of correlation: the same individual risk, but a much fatter chance of both bonds falling over in the same crisis.

Worked example 2 — a shared shock

Imagine 100 bonds, each with a 5% default probability and $\rho = 0.3$ to a shared factor $M$. Draw $M = 0$ (an average day): each bond's threshold is unaffected, so close to 5 out of 100 default. Draw $M = -3$ (a severe systemic shock): the shared term $\rho M = 0.3 \times (-3) = -0.9$ pulls every bond's asset value down at once, so far more than 5 cross below their thresholds — a cluster of defaults arriving together, not because any single company worsened, but because the string was yanked. That is the entire point of the model: it turns "what if the whole economy turns" into a number.

Drag the correlation slider here and watch the scatter tighten around the diagonal — that's the same $\rho$ from the formula above, except instead of two stock returns it is two companies' hidden asset values, and instead of a regression line, what matters is how many points fall in the bottom-left corner (both below their default thresholds) at once.

What this means in practice

This machinery is what prices CDO tranches and credit index tranches: given a portfolio's marginal default curve and a correlation assumption, it produces the loss distribution needed to price a slice that only absorbs losses between, say, 3% and 7% of the pool. Because the model reduces every messy real-world channel of contagion — shared lenders, shared suppliers, panic — down to one number $\rho$, dealers don't guess $\rho$; they back it out from the market prices of tranches, producing a "correlation skew" the same way option markets produce a volatility skew.

Practice

1. Two bonds each have a 10% five-year default probability. Would you expect the joint default probability at $\rho = 0.6$ to be closer to the independent case (1%) or to the single-bond case (10%)? Why?
2. If a systemic shock pulls the shared factor $M$ very negative, does the model predict that low-$\rho$ or high-$\rho$ names in the basket default first?