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# Vasicek Model for Probability of Default Modelling

## What is Vasicek Model?

Bank regulators utilise the Vasicek model to predict the extreme percentile of the loss distribution.

The Vasicek model relates the probability of default to the value of a factor. That factor can be considered a measure of the recent health of the economy. It uses the Gaussian copula model to define the correlation between defaults.

## Example of Vasicek Model:

Assume the probability of default (PD) is the same for all companies in an extensive portfolio. The binary probability of the default distribution for the company i for one year is mapped to a standard normal distribution U, as described in the previous section. Values in the extreme left tail of this standard normal distribution correspond to default, whereas the rest of the distribution corresponds to no default.

Ui ≤ N-1 (PD)

Where N-1 is the inverse cumulative normal distribution.

For example, if PD = 1%, company/defaults if:

Ui ≤ N-1 (0.01) = – 2.326

Values of i, between minus infinity and -2.326 correspond to default, while values between -2.326 and infinity correspond to no default.

## Why is Vasicek Model important?

The Vasicek model exhibits a mean-reversion, which helps predict future interest rate movements. As shown in the table below, when market shocks cause the interest rate (or “short rate”) to be higher than the long term mean, the drift factor (drt = a(b-rt)) is lower than 0 – indicating that the interest rate is likely to decrease.

 Short Rate, rt Rate Change, drt rt > b Short Rate is greater than Long Term Rate drt = a(b – rt) < 0 Adjust Short Rate Process Downwards rt < b Short Rate is less than Long Term Rate drt = a(b – rt) > 0 Adjust Short Rate Process Upwards

Table 1: Mean Reverting Drift: drt = a(b – rt)

Owais Siddiqui