The goal of modelling credit risk is to determine the credit loss distribution. A credit loss is a loss due to debtors who fail to meet their payment obligations in one year. The distribution is a combination of probabilities and losses. For instance:

There is a probability of 2% for a credit loss of €50,000.- or less.

There is a probability of 7% for a credit loss of €100,000.- or less.

There is a probability of 16% for a credit loss of €150,000.- or less.

There is a probability of 31% for a credit loss of €200,000.- or less.

Etc.

These probabilities continue to grow until it is one. The probability of an endless credit loss or less is one. This is because all credit losses are endless or less.

If you have enough estimates of a probability for a loss of X or less a graph can be drawn. Such a graph is called a cumulative distribution function (CDF). The following example shows ten combinations of probabilities and credit loss. Each probability indicates the probability of the associated credit loss or less.

Probability | Credit loss (or less) |

2% | € 50,000 |

7% | € 100,000 |

16% | € 150,000 |

31% | € 200,000 |

50% | € 250,000 |

69% | € 300,000 |

84% | € 350,000 |

93% | € 400,000 |

98% | € 450,000 |

99% | € 500,000 |

These probabilities and associated losses translate to the following CDF:

The CDF graph shows the probability for each level of loss or less. This graph can be transformed into a distribution function by taking its derivative. This strips the “Cumulative” from the CDF. In our example the distribution function would look like this:

The first graph shows the probability of a certain loss or less. The second graph shows the probability of a specific loss. A point on the first graph (the CDF) can be reconstructed from the second graph by adding all the probabilities equal to, or less than, the credit loss. In other words, a point on the first graph (the CDF) can be reconstructed from the loss distribution (the second graph) by taking the area under the graph up to and including the credit loss for which you wish to recreate the point.

The red area in the graph below represents the probability of €200,000 or less. The total value of this area is 31%, which is equal to the probability of €200,000 or less on the CDF graph.

In these examples a normal distribution is used. The actual loss distribution can have any form.

**Translating a loss distribution to capital**

So why is this loss distribution so important? The purpose of holding capital is to ensure that a bank is capable of absorbing loss in an extreme situation. Basel defines an extreme situation as the point on the loss distribution with 99.9% probability of the associated credit loss or less. This is represented by the green and red area of the next graph.

The green and red area combined make up 99.9% of the total area under the graph. In our example the point which represents the 99.9% probability is €559,025. This means there is a probability of 99.9% that the credit risk will be is €559,025 or less.

The green area in the graph represents 50% of the total area under the graph. In other words there is a 50% probability of this loss or less. This level of loss is called the expected loss. In our example the expected loss is €250,000. In other words there is a 50% probability for a loss of €250,000 or less. This part of the risk of loss should be covered by the provisions of a bank. This means that the banks provisions for credit risk should equal the expected loss.

The red area is the remainder between the expected loss and the loss at the 99.9% point. This remainder is called the unexpected loss. The regulatory capital is used to cover the unexpected loss. In our example we showed that the 99.9% point was at a loss of €559,025 and the expected loss was €250,000. Therefore the unexpected loss in our example is €559,025 - €250,000 = €309,025.

The sum of the provisions and the regulatory capital should equal the 99.9% loss. If for some reason the provisions are less than the expected loss, it should be compensated by holding extra regulatory capital.

In the next section we will discuss the method by which Basel attempts to model the credit loss distribution.