{"id":12753,"date":"2026-06-07T05:28:38","date_gmt":"2026-06-07T01:28:38","guid":{"rendered":"https:\/\/eoninvestment.com\/ka\/?p=12753"},"modified":"2026-06-07T06:19:58","modified_gmt":"2026-06-07T02:19:58","slug":"engaussian-copula-simulationkagaussian-copula-%e1%83%a1-%e1%83%a1%e1%83%98%e1%83%9b%e1%83%a3%e1%83%9a%e1%83%90%e1%83%aa%e1%83%98%e1%83%90","status":"publish","type":"post","link":"https:\/\/eoninvestment.com\/ka\/2026\/06\/07\/engaussian-copula-simulationkagaussian-copula-%e1%83%a1-%e1%83%a1%e1%83%98%e1%83%9b%e1%83%a3%e1%83%9a%e1%83%90%e1%83%aa%e1%83%98%e1%83%90\/","title":{"rendered":"Gaussian Copula-\u10e1 \u10e1\u10d8\u10db\u10e3\u10da\u10d0\u10ea\u10d8\u10d0:<\/span>"},"content":{"rendered":"

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<\/strong> for each company and defines correlation between these scores.<\/p>\n\n\n\n

Now let’s see how this model can be used in practice through simulation.<\/p>\n\n\n\n

The main question is:<\/p>\n\n\n\n

How can we estimate how many companies may default at the same time?<\/strong><\/p>\n\n\n\n

1. Starting with Default Probabilities<\/h2>\n\n\n\n

Suppose we have 10 companies.<\/p>\n\n\n\n

Using credit ratings, historical statistics, or market data, we obtain the following cumulative default probabilities:<\/p>\n\n\n\n

\u10ec\u10d4\u10da\u10d8<\/th>Cumulative Default Probability<\/th><\/tr><\/thead>
1<\/td>1%<\/td><\/tr>
2<\/td>3%<\/td><\/tr>
3<\/td>6%<\/td><\/tr>
4<\/td>10%<\/td><\/tr>
5<\/td>15%<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

This table shows the probability that a company has already defaulted by the end of a given year.<\/p>\n\n\n\n

For example, a 15% probability means that there is a 15% chance that the company will default within five years.<\/p>\n\n\n\n

2. Transforming Probabilities into Thresholds<\/h2>\n\n\n\n

The Gaussian Copula converts these probabilities into thresholds on the standard normal distribution.<\/p>\n\n\n\n

In other words, it finds the corresponding number of standard deviations on the familiar bell-shaped curve.<\/p>\n\n\n\n

(For a more detailed explanation, see: Gaussian Copula \u2014 Modeling Default Correlation<\/em>.)<\/a><\/p>\n\n\n\n

The resulting table looks as follows:<\/p>\n\n\n\n

\u10ec\u10d4\u10da\u10d8<\/th>Cumulative Default Probability<\/th>Threshold<\/th><\/tr><\/thead>
1<\/td>1%<\/td>-2.33<\/td><\/tr>
2<\/td>3%<\/td>-1.88<\/td><\/tr>
3<\/td>6%<\/td>-1.55<\/td><\/tr>
4<\/td>10%<\/td>-1.28<\/td><\/tr>
5<\/td>15%<\/td>-1.04<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

The lower the Hidden Credit Score, the worse the company’s financial condition.<\/p>\n\n\n\n

In practical terms:<\/p>\n\n\n\n

If a company’s Hidden Credit Score falls below -1.04, it will default within five years.<\/strong><\/p>\n\n\n\n

3. Introducing Correlation<\/h2>\n\n\n\n

In reality, defaults are correlated due to macroeconomic events and interconnected business relationships.<\/p>\n\n\n\n

To capture this effect, the model introduces a correlation assumption.<\/p>\n\n\n\n

In our example:<\/p>\n\n\n\n

p = 20%<\/p>\n\n\n\n

Conceptually, this means that the Hidden Credit Scores of different companies are correlated by 20%.<\/p>\n\n\n\n

\u10db\u10d0\u10e0\u10e2\u10d8\u10d5\u10d0\u10d3:<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

4. Splitting Risk into Two Components<\/h2>\n\n\n\n

The model assumes that each company’s Hidden Credit Score consists of two components:<\/p>\n\n\n\n

x<\/mi>i<\/mi><\/msub>=<\/mo>\u03c1<\/mi><\/msqrt>F<\/mi>+<\/mo>1<\/mn>\u2212<\/mo>\u03c1<\/mi><\/mrow><\/msqrt>Z<\/mi>i<\/mi><\/msub><\/mrow>x_i = \\sqrt{\\rho}F + \\sqrt{1-\\rho}Z_i<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

where:<\/p>\n\n\n\n