# Modification of the Vasicek Model

We propose one of possible implementations of stress-testing for the credit portfo­lios of corporate borrowers (further—the Model), which is based on Monte-Carlo simulations and the modified Vasicek model (Vasicek 1987). As will be shown, the Model meets all of the criteria described above.

The single systemic factor Vasicek model is based on the assumption that assets of the companies have two drivers—idiosyncratic (determining the individual prop­erties of each company) and systemic (the overall macroeconomic environment). The change in assets of company Ai, according to the Vasicek model, is equal to the sum of two normally distributed random variables: idiosyncratic – ei and systematic Z; the level of the dependence of the borrower from the systematic factor is captured by the correlation coefficient pi:

If the value of company assets becomes less than some threshold level default occurs. Usually, the default threshold is defined as a company’s debt burden. The default threshold could be calibrated based on the assumption of the normal distribution of asset return values A; and a given borrower’s default probability:

P (A; < ThsO = PD,- => N (ThsO = PD; => Ths; = N_1 (PDO

We propose the following modification of the Vasicek model for stress-testing pur­poses: The default threshold should be decomposed on the sum of the components, each component consisting of the macro-variable Mj multiplied by coefficient, which defines the degree of dependence between default frequency and macro­variable j.

m

Thsi = a, + £Pij • Mj (2)

j =1

The proposed threshold decomposition will allow us to capture historical depen­dence between defaults and macro-variables, which also serves as a default corre­lation transmitter due to asset value dependence on the same factors. At the same time, the model contains explicit default correlation parameter p;, by which we take into account the default correlation, which is not detectable through dependence on macro factors.

It is impossible to statistically identify dependence between macro-variables and individual borrowers; therefore companies should be grouped into subsets with similar risk characteristics—rating classes.

It is very important to choose macro-factors for model calibration correctly. As general recommendations, we propose the following selection criteria:

1. Each macro-variable should have significant individual predictive power regard­ing historical default frequencies (R2).

2. The correlation between selected macro-variables should be relatively low (for purposes of model stability).

A high correlation between all predictive macro-variables is common for emerg­ing economies (for example, the price of oil in OPEC countries or Russia is the main economic driving force); therefore in order to fulfill the second requirement it is recommended to replace the original dynamics of the macro-variables Mi by the

principle components of macro-variables Mt with zero correlation between them.

According to the proposed modifications, the density function for default frequency could be written as:

 Em ■ * Mqj + J~Pi ‘ Z J-1 1 – N V

 (3)  where

Q—index of the time period (quarter or year).

nq—number of borrowers in the portfolio during the period q.

xq—number of defaults during the period q.

N—normal distribution function.

Mj—historical value of j macro-variable during the period q.

R—number of rating classes.

Given the default density function, information of the historical default frequen­cies by rating classes and historical values of macro-variables, parameters at, ,

Pi could be found using the maximum likelihood approach. As a result, we could produce conditional PDs for rating classes given the macro-forecast.