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MONTE CARLO SIMULATIONS OF DEFAULT EVENTS BASED ON THE STRUCTURAL MODEL OF DEFAULT

The example uses the standardized normal distribution of distance to default model, given the default probability of obligors. The model operates in default mode only, and exposures are at book value. There is a uniform preset asset correlation and a uniform preset default probability.

Asset Distribution and Default Probability

We use the standardized normal distance to default model, considering that the default probability embeds all information relevant to default and on unobservable asset values. Considering a single firm "i" with the default probability a, there is a default point ^(a) of the random asset value A. such that the default probability is a. If O is the cumulated standardized normal distribution, then:

<E>~' is the normal inverse distribution. The normal inverse distribution provides the given threshold^(a) given the default probability a. For instance, with a 1%, the default point is A(a{) — <&_1(1%) -2.33, and so on. Generating a default event requires that the asset value A. be lower than^(a) = ^'(CX;).

A dummy variable represents the default event. It has a value of 1 if the asset value is below the threshold (default), and zero otherwise (non-default):

The dummy variables are intermediate variables serving for determining the portfolio number of defaults.

The Multiple Simulations

There are N obligors in the portfolio. The number of simulations of all asset values of all obligors is K. Each simulated asset value determines a 1 or 0 value for each d., one for each of the N obligors, with / varying from 1 to N. Each simulation generates as many asset values as there are obligors, or N asset values. For each obligor, the discrete default variable d. takes the values 0 or 1. The sum of all d. is the portfolio number of defaults for one simulation.

The simulations are repeated Crimes. For each simulation, we have TV asset values, TV values for the dummy variable, and a total count of defaults. With K simulations, we have K numbers of portfolio defaults. The K counts of portfolio defaults form a distribution. With a large number of simulations, we have a smoother distribution. From the distribution of the number of portfolio defaults, all statistics, standard deviations of defaults, mode of the distribution, and percentiles of the count of portfolio defaults are derived.

 
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