The Vasicek Value at Risk model is a special case of the Gaussian Copula model.
To recall, in the previous posts we started with a simple idea: every company has a hidden credit score. If this hidden score deteriorates beyond a certain threshold, the company defaults.
Then we saw that the hidden credit score consists of two factors: one common factor shared by the companies in the portfolio, and one company-specific factor. When the common factor deteriorates, many companies can default together. This is the logic behind modeling default correlation.
After that, in the Excel model, we used Monte Carlo simulation to show how correlation between defaults works in practice through the Gaussian Copula simulation.
Vasicek Credit VaR shows how default correlation works for a large homogeneous portfolio — for example, a portfolio consisting of 1,000+ similar credit instruments.
The interesting point is that in the Excel model, simulation is no longer required. Vasicek transforms the model into an algebraic formula, because a large number of similar exposures allows the specific-risk component to be diversified away.
What Credit VaR Measures
Credit VaR asks a simple question:
How large can credit losses become in a bad scenario?
For example, a 1-year 99.9% Credit VaR of $5.13 million means that, over one year, with 99.9% confidence, the maximum credit loss is $5.13 million.
There is a difference between Market VaR and Credit VaR.
The difference is the source of the loss.
In Market VaR, the question is:
How much can we lose because the market prices of securities change?
In Credit VaR, the question is:
How much can we lose because borrowers default?
Vasicek’s Default Formula
The formula is:
Where:
- V(X,T) – is the default rate in the bad scenario;
- Q(T) – is the cumulative probability of default over time (T);
- p – is the correlation between hidden credit scores;
- X – is the confidence level.
After we calculate V(X,T) , Credit VaR is:
Where:
- L – is the size of the portfolio;
- R – is the recovery rate.
Hull’s Example in Excel
To summarize, the model looks like this:
The interesting point is that if events develop according to the worst 0.1% scenario, the loss is 6.4 times larger than the average expected loss, as shown in the final line.
Sensitivity Analysis
Now let us look at how three different factors affect the expected loss in a bad scenario.
1. Impact of changing the correlation rate
2. Impact of cumulative default probability / credit rating
3. Impact of the confidence level
Excel — Vasicek VaR Model
Adapted from:
Options, Futures, and Other Derivatives — John C. Hull
