Monte-Carlo Simulation Schema
One of the key requirements for the stress-testing model is the ability to estimate changes in the rating structure of the portfolio over time. The most obvious approach for this task is to incorporate migration matrixes into the model. Due to the dependence of the rating migration dynamics on the economic cycle, it is recommended to use different migration matrixes for stress and expansion scenarios.
We propose the following Monte-Carlo simulation schema, which takes into account the proposed density function (3) and migration matrixes:
1. For the given macro-variable dynamics (from the macro-forecast) for the stresstesting period, conditional PDs are calculated [using (2)] for each rating class— Thsi.
2. The normal random variable Z is generated (systemic factor).
3. The normal random variable ei is generated for each borrower in the portfolio (idiosyncratic factor).
4. If ei • V1 — p + Z • ^/p < Ths,-, a default event is fixed for a borrower during a current period. A defaulted borrower is excluded from the portfolio, and its exposure multiplied by LGD is added to the total portfolio losses within the scenario.
5. If the borrower does not default, its ratings for the next period are changed in accordance with the migration matrix—a uniformly distributed random number is generated r 2 [0; 1], and a new rating for the next period is assigned to the borrower according to the probabilistic interval of the migration matrix in which random number r falls.
6. Items 1-5 are repeated until the required forecast horizon is achieved.
The result of MC simulations is an array of losses. This array is a numerical representation of the density function of losses due to borrowers defaulting. On the basis of this distribution, the mean and quantiles of portfolio losses can be estimated.
The marginal contribution of individual borrowers to the stress-test results can be estimated using an approach similar to the Monte-Carlo model, which is described, for example, in (Tasche 2000).