Gaussian Copula Simulation

In the previous article, we discussed the core idea behind the Gaussian Copula model. The model does not correlate defaults directly. Instead, it creates a Hidden Credit Score for each company and defines correlation between these scores.

Now let’s see how this model can be used in practice through simulation.

The main question is:

How can we estimate how many companies may default at the same time?

1. Starting with Default Probabilities

Suppose we have 10 companies.

Using credit ratings, historical statistics, or market data, we obtain the following cumulative default probabilities:

Year	Cumulative Default Probability
1	1%
2	3%
3	6%
4	10%
5	15%

This table shows the probability that a company has already defaulted by the end of a given year.

For example, a 15% probability means that there is a 15% chance that the company will default within five years.

2. Transforming Probabilities into Thresholds

The Gaussian Copula converts these probabilities into thresholds on the standard normal distribution.

In other words, it finds the corresponding number of standard deviations on the familiar bell-shaped curve.

(For a more detailed explanation, see: Gaussian Copula — Modeling Default Correlation.)

The resulting table looks as follows:

Year	Cumulative Default Probability	Threshold
1	1%	-2.33
2	3%	-1.88
3	6%	-1.55
4	10%	-1.28
5	15%	-1.04

The lower the Hidden Credit Score, the worse the company’s financial condition.

In practical terms:

If a company’s Hidden Credit Score falls below -1.04, it will default within five years.

3. Introducing Correlation

In reality, defaults are correlated due to macroeconomic events and interconnected business relationships.

To capture this effect, the model introduces a correlation assumption.

In our example:

p = 20%

Conceptually, this means that the Hidden Credit Scores of different companies are correlated by 20%.

In simple terms:

When the credit condition of one company deteriorates, the credit conditions of other companies tend to deteriorate as well, because they operate within the same economic environment.

4. Splitting Risk into Two Components

The model assumes that each company’s Hidden Credit Score consists of two components:

where:

• x_i = Hidden Credit Score of company (i)
• F = Common economic factor
• Z_i = Company-specific factor
• p = Correlation coefficient

In our example:

x_i = 0.447 F + 0.894 Z_i

In Excel, this means that approximately 20% of the variation in credit quality is driven by the common economic factor, while the remaining 80% comes from company-specific factors.

5. What Are We Simulating?

This is the most important question.

The default probabilities and correlation assumption are already known.

What we simulate are:

• The common economic factor (F)
• The company-specific factors (Z_i)

In other words, we generate many possible future scenarios.

In each scenario:

• The economy may perform well or poorly.
• Individual companies may experience favorable or unfavorable events.
• Together, these factors determine the Hidden Credit Scores.

For example:

F = -1.5

represents a weak economic environment.

Each company then receives its own individual shock:

Z₁, Z₂, …, Z_{10}

Using these values, we calculate:

x₁, x₂, …, x_{10}

One can think of this process as rolling the dice and generating many possible versions of the future.

6. Comparing Scores with Thresholds

Suppose one company receives:

x = – 1.70

We compare this score with the thresholds:

Year	Threshold
1	-2.33
2	-1.88
3	-1.55
4	-1.28
5	-1.04

A value of -1.70 is not low enough for default in Year 1 or Year 2.

However: -1.70 < -1.55, Therefore, the company defaults during Year 3.

After evaluating all companies, we obtain a table showing the default year for each company in a particular simulated scenario. Each row of the simulation represents a different possible future.

7. Counting Defaults

At the end of each simulation, we count how many companies have defaulted by Year 1, 2, 3, 4, 5

This allows us to observe how defaults accumulate across the portfolio.

8. Repeating the Process

In our Excel model, we run 50 simulations.

However, to obtain a smoother and more reliable picture, at least 500–1,000 iterations are usually desirable.

Professional risk models often use tens of thousands of simulations.

9. Reading the Results

The final output for our portfolio looks like a distribution.

The horizontal axis shows the number of defaults.

The vertical axis shows how many simulated scenarios produced that number of defaults.

In other words, among all the simulated futures, the most common outcome is a single default.

As we move to the right, scenarios with larger numbers of defaults become less frequent.

With more simulations, the distribution becomes smoother.

For investors and risk managers, the shape of this distribution is extremely important.

It allows different portfolios to be compared and helps identify the probability of extreme loss events.

Excel File

Gaussian Copula Simulation

P.S.

One important limitation is that the correlation parameter is assumed to be constant.

In practice, many professionals believe that correlations increase during periods of economic stress.

For this reason, more sophisticated versions of the model are often used.

Adapted from:

John C. Hull — Options, Futures, and Other Derivatives